logcondens/0000755000176200001440000000000014131330403012375 5ustar liggesuserslogcondens/NAMESPACE0000644000176200001440000000147012615675116013640 0ustar liggesusers## which functions shall be accessible by the users? export( activeSetLogCon, confIntBootLogConROC_t0, etaphi, evaluateLogConDens, icmaLogCon, intECDF, intF, isoMean, J00, J10, J11, J20, Lhat_eta, Local_LL, Local_LL_all, LocalCoarsen, LocalConvexity, LocalExtend, LocalF, LocalMLE, LocalNormalize, LocalVariance, logConDens, logConROC, logconTwoSample, maxDiffCDF, MLE, phieta, plot.dlc, preProcess, Q00, qloglin, quadDeriv, quantilesLogConDens, rlogcon, robust, ROCx, summary.dlc, logConCI, ftilde, c2hat ) importFrom(ks, hscv, kdde, hlscv, hpi, hns) importFrom(graphics, abline, legend, lines, mtext, par, plot, rug, segments) importFrom(stats, dnorm, integrate, pnorm, quantile, rnorm, runif, uniroot) S3method(summary, dlc, summary.dlc) S3method(plot, dlc, plot.dlc) logcondens/data/0000755000176200001440000000000014131325162013314 5ustar liggesuserslogcondens/data/pancreas.rda0000644000176200001440000000172114131325261015601 0ustar liggesusersKTQǟ? ,I34,1a9;EiX=,#RGE$EFP(-#m"Z.*h*Zdo<9@›{s=״%TJ,+JOh-p0n=b-+!͑slf)s:fYX1;qy|-Ů yߏag)\އ܀ޅ""[r> ^ta4_,䕠әϋkR?^ </@< ɛz="L ."H9sHp׍ `vI9"8 {!"<'$s-$>>LXw P"Ww rf?Gc_#~:c;})c[^)u~~_XrqD$Od~tb}S _29}$ _p[ O rs Z`ď:/~Ʈzb`=5o+O7eM\`~߇\Z7 6~{âOZ[C\̗ ;}l_%H^t\Ưs ~[9_~=qnKZcFzjO&]F}xkܭE6S}i}+^{t-Tˀ/_'=[=y#Ѿ)3@]CG8nSq[":%zXֈgηnlZg&mPCñN;:vu8uq~fff q logcondens/data/reliability.rda0000644000176200001440000000713714131325261016325 0ustar liggesusers]Xy\l٦$u؉dnFNBTdKyYeKH%dd~>y~gyy_}; p`o2lMv66&BW;[T_KL5aܰ&,m2rv4q&hJ]oUzw.kƼ}@ htxnG2G<"vIZO}hQ {mq\Wi#^1lF}~s]r-k|Цb_=D'FsZru~\Ag`|Q#mJ05=p==[{aMgx6}$ntƃwղ:0G*F=Khos{&^~]tO_-\/^ĥ/'+/3zb?F^W1oe}%FnswO!Xy4o-pfsDdxNVzin$E)Ac3!qB2o<~y0aCSLgyX+WCO9}#p'sGXOHz C>喙>|}bpݗȳ"uū)╁<1NěbbO IoIXY 7| ֧1/&w~?#?#`Hq"^X/lm-/.`70?x u> WyD#mDTx%Շ#+~ut D_xaOZ]߇uF74#+"OedDZS\yHґ>}x"ϕ^lF%M)}\v@75Rs+ٔ XЏ߁]Va`UF$Ssy@?*>{'? 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JZfHD4\=vJ AZ7,û!"+SSc\77*J +r&je x=W?AX`Ģ+{UPV"I>Z/}V 'Щٝsv5jbu0a}]g>h{870?}5`/) OŤ2V! (401l@}s&Ytk5v|epVĠhZɬ E|RxgXo.~]9g|+%`f5ajMɃe?>!Xi(0 YZlogcondens/man/0000755000176200001440000000000014131325164013160 5ustar liggesuserslogcondens/man/reliability.Rd0000644000176200001440000000173614130607006015764 0ustar liggesusers\name{reliability} \alias{reliability} \docType{data} \title{Reliability dataset used to illustrate log-concave density estimation} \description{Dataset that contains the data analyzed in Duembgen and Rufibach (2009, Figure 2).} \usage{data(reliability)} \format{A vector containing the 786 observations analyzed in Duembgen and Rufibach (2009, Figure 2).} \source{The data was taken from Duembgen and Rufibach (2009).} \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \seealso{See the example in \code{\link{logConDens}} for the analysis of this data.} \keyword{datasets} logcondens/man/logConCIfunctions.Rd0000644000176200001440000000162614131322367017044 0ustar liggesusers\name{logConCIfunctions} \alias{logConCIfunctions} \alias{ftilde} \alias{c2hat} \alias{rLCD} \title{Functions that are used by logConCI} \description{Functions that are used by \code{\link{logConCI}} and are not intended to be called by the user.} \references{ Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities. \emph{Comput. Statist. Data Anal.}, \bold{75}, 248--264. Baladbaoui, F., Rufibach, K. and Wellner, J. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. \emph{Ann. Statist.}, \bold{37(3)}, 1299--1331. Tarn Duong (2012). ks: Kernel smoothing. R package version 1.8.10. \url{https://CRAN.R-project.org/package=ks} } \author{ Mahdis Azadbakhsh Hanna Jankowski, \email{hkj@mathstat.yorku.ca}, \cr \url{http://www.math.yorku.ca/~hkj/} } \keyword{htest} \keyword{nonparametric} logcondens/man/logConROC.Rd0000644000176200001440000001045214130606724015241 0ustar liggesusers\name{logConROC} \alias{logConROC} \title{Compute ROC curve based on log-concave estimates for the constituent distributions} \description{The receiver operating characteristic (ROC) curve for two constituent distributions \eqn{F} and \eqn{G} is defined as \deqn{R(t; F, G) = 1 - G(F^{-1}(1 - t))} for \eqn{t \in [0, 1]}. It is typically used to assess the performance of a diagnostic test used to discriminate between healthy and diseased individuals based on a continuous variable. } \usage{logConROC(cases, controls, grid, smooth = TRUE)} \arguments{ \item{cases}{A vector of measurements for the cases.} \item{controls}{A vector of measurements for the controls.} \item{grid}{A vector specifying the grid where the ROC curve is computed on.} \item{smooth}{Logical, indicating whether ROC curve and AUC should also be computed based on the smoothed log-concave density estimator.} } \details{In Rufibach (2011) it was shown that the ROC curve based on log-concave density estimates exhibit nice properties for finite sample sizes as well as asymptotically. Its performance is typically much better than that of the empirical ROC curve and only, if at all, sligthly worse compared to the binormal model when in fact the underlying densities are normal. However, log-concavity encompasses many parametric densities, so this new model is much more flexible than the binormal one, at little efficiency sacrifice.} \value{ \item{m}{Number of control measurements.} \item{n}{Number of case measurements.} \item{fROC}{Estimated ROC curve based on the log-concave density estimate.} \item{fROC.smooth}{Estimated ROC curve based on the smoothed log-concave density estimate.} \item{res0}{\code{dlc} object as a result of a call to \code{\link{logConDens}} for the data of the controls.} \item{res1}{\code{dlc} object as a result of a call to \code{\link{logConDens}} for the data of the cases.} } \references{ Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011). logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. \emph{Int. J. Biostat.}, \bold{8}(1), 1--29. } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch}} \seealso{Confidence intervals at given false-positive fractions for the ROC curve based on log-concave densities can be computed using \code{\link{confIntBootLogConROC_t0}}. For the computation of the AUC the function \code{\link{ROCx}} is used. In the example below we analyze the \code{\link{pancreas}} data.} \examples{ ## ROC curve for pancreas data data(pancreas) status <- factor(pancreas[, "status"], levels = 0:1, labels = c("healthy", "diseased")) var <- log(pancreas[, "ca199"]) cases <- var[status == "diseased"] controls <- var[status == "healthy"] ## compute and plot empirical ROC curve ## code modified from https://stat.ethz.ch/pipermail/r-help/2008-October/178531.html xx <- c(-Inf, sort(unique(c(cases, controls))), Inf) sens <- sapply(xx, function(x){mean(cases >= x)}) spec <- sapply(xx, function(x){mean(controls < x)}) ## compute log-concave ROC curve grid <- seq(0, 1, by = 1 / 500) roc.logcon <- logConROC(cases, controls, grid) ## plot plot(0, 0, xlim = c(0, 1), ylim = c(0, 1), type = 'l', main = "ROC curves for pancreas data", xlab = "1 - specificity", ylab = "sensitivity", pty = 's') legend("bottomright", c("empirical ROC", "log-concave ROC", "smooth log-concave ROC"), lty = c(1, 1, 2), lwd = 2, col = 2:4, bty = "n") segments(0, 0, 1, 1, col = 1) lines(1 - spec, sens, type = 'l', col = 2, lwd = 2) lines(grid, roc.logcon$fROC, col = 3, lwd = 2) lines(grid, roc.logcon$fROC.smooth, col = 4, lwd = 2, lty = 2) \dontrun{ ## bootstrap confidence intervals at 1 - specificity = 0.2 and 0.8: res <- confIntBootLogConROC_t0(controls, cases, grid = c(0.2, 0.8), conf.level = 0.95, M = 1000, smooth = TRUE, output = TRUE) res } } \keyword{nonparametric} logcondens/man/pancreas.Rd0000644000176200001440000000271412615675116015261 0ustar liggesusers\name{pancreas} \alias{pancreas} \docType{data} \title{Data from pancreatic cancer serum biomarker study} \description{First published by Wieand et al (1989), this dataset contains data on serum measurements for a cancer antigen (CA-125) and a carbohydrate antigen (CA19.9) both of which are measured on a continuous non-negative scale. The measurements were taken within a case-control study on 90 cases with pancreatic cancer and 51 controls who did not have cancer but pancreatitis. The primary question of the study was which one of the two biomarkers best distinguishes cases from controls.} \usage{data(pancreas)} \format{ A data frame with 141 observations on the following 3 variables. \describe{ \item{\code{ca199}}{CA19.9 measurements.} \item{\code{ca125}}{CA125 measurements.} \item{\code{status}}{Patient status, 0 for controls and 1 for cases.} } } \source{The data was downloaded from http://labs.fhcrc.org/pepe/book/ on February 2, 2011.} \references{ Wieand, S., Gail, M. H., James, B. R., and James, K.L. (1989). A family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data. \emph{Biometrika}, \bold{76}, 585--592. Pepe, M.S. (2003) \emph{The statistical evaluation of medical tests for classification and prediction}. Oxford: Oxford University Press (Section 1.3.3). } \seealso{This data is analyzed in the help file for the function \code{\link{logConROC}}.} \keyword{datasets} logcondens/man/logConDens.Rd0000644000176200001440000001303714130622661015507 0ustar liggesusers\name{logConDens} \alias{logConDens} \title{Compute log-concave density estimator and related quantities} \description{Compute the log-concave and smoothed log-concave density estimator.} \usage{logConDens(x, xgrid = NULL, smoothed = TRUE, print = FALSE, gam = NULL, xs = NULL)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, not necessarily unique.} \item{xgrid}{Governs the generation of weights for observations. See \code{\link{preProcess}} for details.} \item{smoothed}{If \code{TRUE}, the smoothed version of the log-concave density estimator is also computed.} \item{print}{\code{print = TRUE} outputs the log-likelihood in every loop, \code{print = FALSE} does not. Make sure to tell \code{R} to output (press CTRL+W).} \item{gam}{Only necessary if \code{smoothed = TRUE}. The standard deviation of the normal kernel. If equal to \code{NULL}, \code{gam} is chosen such that the variances of the original sample \eqn{x_1, \ldots, x_n} and \eqn{\widehat f_n^*}{\hat f_n^*} coincide.} \item{xs}{Only necessary if \code{smoothed = TRUE}. Either provide a vector of support points where the smoothed estimator should be computed at, or leave as \code{NULL}. Then, a sufficiently width equidistant grid of points will be used.} } \details{See \code{\link{activeSetLogCon}} for details on the computations.} \value{ \code{\link{logConDens}} returns an object of class \code{"dlc"}, a list containing the following components: \code{xn}, \code{x}, \code{w}, \code{phi}, \code{IsKnot}, \code{L}, \code{Fhat}, \code{H}, \code{n}, \code{m}, \code{knots}, \code{mode}, and \code{sig} as generated by \code{\link{activeSetLogCon}}. If \code{smoothed = TRUE}, then the returned object additionally contains \code{f.smoothed}, \code{F.smoothed}, \code{gam}, and \code{xs} as generated by \code{\link{evaluateLogConDens}}. Finally, the entry \code{smoothed} of type \code{"logical"} returnes the value of \code{smoothed}. The methods \code{\link{summary.dlc}} and \code{\link{plot.dlc}} are used to obtain a summary and generate plots of the estimated density. } \references{ Duembgen, L, Huesler, A. and Rufibach, K. (2010). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}. Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011). logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \examples{ ## =================================================== ## Illustrate on simulated data ## =================================================== ## Set parameters n <- 50 x <- rnorm(n) res <- logConDens(x, smoothed = TRUE, print = FALSE, gam = NULL, xs = NULL) summary(res) plot(res, which = "density", legend.pos = "topright") plot(res, which = "log-density") plot(res, which = "CDF") ## Compute slopes and intercepts of the linear functions that ## compose phi slopes <- diff(res$phi) / diff(res$x) intercepts <- -slopes * res$x[-n] + res$phi[-n] ## =================================================== ## Illustrate method on reliability data ## Reproduce Fig. 2 in Duembgen & Rufibach (2009) ## =================================================== ## Set parameters data(reliability) x <- reliability n <- length(x) res <- logConDens(x, smooth = TRUE, print = TRUE) phi <- res$phi f <- exp(phi) ## smoothed log-concave PDF f.smoothed <- res$f.smoothed xs <- res$xs ## compute kernel density sig <- sd(x) h <- sig / sqrt(n) f.kernel <- rep(NA, length(xs)) for (i in 1:length(xs)){ xi <- xs[i] f.kernel[i] <- mean(dnorm(xi, mean = x, sd = h)) } ## compute normal density mu <- mean(x) f.normal <- dnorm(xs, mean = mu, sd = sig) ## =================================================== ## Plot resulting densities, i.e. reproduce Fig. 2 ## in Duembgen and Rufibach (2009) ## =================================================== plot(0, 0, type = 'n', xlim = range(xs), ylim = c(0, 6.5 * 10^-3)) rug(res$x) lines(res$x, f, col = 2) lines(xs, f.normal, col = 3) lines(xs, f.kernel, col = 4) lines(xs, f.smoothed, lwd = 3, col = 5) legend("topleft", c("log-concave", "normal", "kernel", "log-concave smoothed"), lty = 1, col = 2:5, bty = "n") ## =================================================== ## Plot log-densities ## =================================================== plot(0, 0, type = 'n', xlim = range(xs), ylim = c(-20, -5)) legend("bottomright", c("log-concave", "normal", "kernel", "log-concave smoothed"), lty = 1, col = 2:5, bty = "n") rug(res$x) lines(res$x, phi, col = 2) lines(xs, log(f.normal), col = 3) lines(xs, log(f.kernel), col = 4) lines(xs, log(f.smoothed), lwd = 3, col = 5) ## =================================================== ## Confidence intervals at a fixed point for the density ## see help file for logConCI() ## =================================================== } \keyword{htest} \keyword{nonparametric} logcondens/man/logConCI.Rd0000644000176200001440000001577014131322204015106 0ustar liggesusers\name{logConCI} \alias{logConCI} \title{Compute pointwise confidence interval for a density assuming log-concavity} \description{Compute approximate confidence interval for the true log-concave density, on a grid of points. Two main approaches are implemented: In the first, the confidence interval at a fixed point is based on the pointwise asymptotic theory for the log-concave maximum likelihood estimator (MLE) developed in Balabdaoui, Rufibach, and Wellner (2009). In the second, the confidence interval is estimated via the boostrap.} \usage{logConCI(res, xx0, conf.level = c(0.8, 0.9, 0.95, 0.99)[3], type = c("DR", "ks", "nrd", "ECDFboot", "NPMLboot")[2], htype = c("hscv", "hlscv", "hpi", "hns")[4], BB = 500)} \arguments{ \item{res}{An object of class \code{dlc}, usually a result of a call to \code{logConDens}.} \item{xx0}{Vector of grid points at which to calculate the confidence interval.} \item{conf.level}{Confidence level for the confidence interval(s). The default is 95\%.} \item{type}{Vector of strings indicating type of confidence interval to compute. When \code{type = ks} is chosen, then \code{htype} should also be specified. The default is \code{type = ks}.} \item{htype}{Vector of strings indicating bandwidth selection method if \code{type = ks}. The default is \code{htype = hns}.} \item{BB}{number of iterations in the bootstrap if \code{type = NPMLboot} or \code{type = ECDFboot}. The default is \code{BB = 500}.} } \value{The function returns a list containing the following elements: \item{fhat}{MLE evaluated at grid points.} \item{up_DR}{Upper confidence interval limit when \code{type = DR}.} \item{lo_DR}{Lower confidence interval limit when \code{type = DR}.} \item{up_ks_hscv}{Upper confidence interval limit when \code{type = ks} and \code{htype = hscv}.} \item{lo_ks_hscv}{Lower confidence interval limit when \code{type = ks} and \code{htype = hscv}.} \item{up_ks_hlscv}{Upper confidence interval limit when \code{type = ks} and \code{htype = hlscv}.} \item{lo_ks_hlscv}{Lower confidence interval limit when \code{type = ks} and \code{htype = hlscv}.} \item{up_ks_hpi}{Upper confidence interval limit when \code{type = ks} and \code{htype = hpi}.} \item{lo_ks_hpi}{Lower confidence interval limit when \code{type = ks} and \code{htype = hpi}.} \item{up_ks_hns}{Upper confidence interval limit when \code{type = ks} and \code{htype = hns}.} \item{lo_ks_hns}{Lower confidence interval limit when \code{type = ks} and \code{htype = hns}.} \item{up_nrd}{Upper confidence interval limit when \code{type = nrd}.} \item{lo_nrd}{Lower confidence interval limit when \code{type = nrd}.} \item{up_npml}{Upper confidence interval limit when \code{type = NPMLboot}.} \item{lo_npml}{Lower confidence interval limit when \code{boot = NPMLboot}.} \item{up_ecdf}{Upper confidence interval limit when \code{boot = ECDFboot}.} \item{lo_ecdf}{Lower confidence interval limit when \code{boot = ECDFboot}.} } \details{In Balabdaoui et al. (2009) it is shown that (if the true density is strictly log-concave) the limiting distribution of the MLE of a log-concave density \eqn{\widehat f_n} at a point \eqn{x} is \deqn{n^{2/5}(\widehat f_n(x)-f(x)) \to c_2(x) \bar{C}(0).} The nuisance parameter \eqn{c_2(x)} depends on the true density \eqn{f} and the second derivative of its logarithm. The limiting process \eqn{\bar{C}(0)} is found as the second derivative at zero of a particular operator (called the "envelope") of an integrated Brownian motion plus \eqn{t^4}. Three of the confidence intervals are based on inverting the above limit using estimated quantiles of \eqn{\bar{C}(0)}, and estimating the nuisance parameter \eqn{c_2(x)}. The options for the function \code{logConCI} provide different ways to estimate this nuisance parameter. If \code{type = "DR"}, \eqn{c_2(x)} is estimated using derivatives of the smoothed MLE as calculated by the function \code{logConDens} (this method does not perform well in simulations and is therefore not recommended). If \code{type="ks"}, \eqn{c_2(x)} is estimated using kernel density estimates of the true density and its first and second derivatives. This is done using the \code{R} package \pkg{ks}, and, with this option, a bandwidth selection method \code{htype} must also be chosen. The choices in \code{htype} correspond to the various options for bandwidth selection available in \pkg{ks}. If \code{type = "nrd"}, the second derivative of the logarithm of the true density in \eqn{c_2(x)} is estimated assuming a normal reference distribution. Two of the confidence intervals are based on the bootstrap. For \code{type = "ECDFboot"} confidence intervals based on re-sampling from the empirical cumulative distribution function are computed. For \code{type = "NPMLboot"} confidence intervals based on re-sampling from the nonparametric maximum likelihood estimate of log-concave density are computed. Bootstrap confidence intervals take a few minutes to compute! The default option is \code{type = "ks"} with \code{htype = "hns"}. Currently available confidence levels are 80\%, 90\%, 95\% and 99\%, with a default of 95\%. Azadbakhsh et al. (2014) provides an empirical study of the relative performance of the various approaches available in this function. } \references{ Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities. \emph{Comput. Statist. Data Anal.}, \bold{75}, 248--264. Baladbaoui, F., Rufibach, K. and Wellner, J. (2009) Limit distribution theory for maximum likelihood estimation of a log-concave density. \emph{Ann. Statist.}, \bold{37(3)}, 1299--1331. Tarn Duong (2012). ks: Kernel smoothing. R package version 1.8.10. \url{https://CRAN.R-project.org/package=ks} } \author{ Mahdis Azadbakhsh Hanna Jankowski, \email{hkj@mathstat.yorku.ca}, \cr \url{http://www.math.yorku.ca/~hkj/} } \examples{ \dontrun{ ## =================================================== ## Confidence intervals at a fixed point for the density ## =================================================== data(reliability) x.rel <- sort(reliability) # calculate 95% confidence interval(s) and plot the result grid <- seq(min(x.rel), max(x.rel), length.out = 200) res <- logConDens(x.rel) ci <- logConCI(res, grid, type = c("nrd", "ECDFboot")) par(las = 1, mar = c(2.5, 3.5, 0.5, 0.5)) hist(x.rel, n = 25, col = gray(0.9), main = "", freq = FALSE, xlab = "", ylab = "", ylim = c(0, 0.0065), border = gray(0.5)) lines(grid, ci$fhat, col = "black", lwd = 2) lines(grid, ci$lo_nrd, col = "red", lwd = 2, lty = 2) lines(grid, ci$up_nrd, col = "red", lwd = 2, lty = 2) lines(grid, ci$lo_ecdf, col = "blue", lwd = 2, lty = 2) lines(grid, ci$up_ecdf, col = "blue", lwd = 2, lty = 2) legend("topleft", col = c("black", "blue", "red"), lwd = 2, lty = c(1, 2, 2), legend = c("log-concave NPMLE", "CI for type = nrd", "CI for type = ECDFboot"), bty = "n") }} \keyword{htest} \keyword{nonparametric} logcondens/man/intECDF.Rd0000644000176200001440000000477114130622661014675 0ustar liggesusers\name{intECDF} \alias{intECDF} \title{Computes the Integrated Empirical Distribution Function at Arbitrary Real Numbers in s} \description{Computes the value of \deqn{\bar{I}(t) = \int_{x_1}^t \bar{F}(r) d \, r}{\bar{I}(t) = \int_{x_1}^t \bar{F}(r) dr} where \eqn{\bar F} is the empirical distribution function of \eqn{x_1,\ldots,x_m}, at all real numbers \eqn{t} in the vector \eqn{\bold{s}}{s}. Note that \eqn{t} (so all elements in \eqn{\bold{s}}{s}) must lie in \eqn{[x_1,x_m]}. The exact formula for \eqn{\bar I(t)} is \deqn{\bar I(t) = \Big(\sum_{i=2}^{i_0}(x_i-x_{i-1})\frac{i-1}{n} \Big) + (t-x_{i_0})\frac{i_0-1}{n}}{\bar I(t) = (\sum_{i=2}^{i_0}(x_i-x_{i-1}) (i-1)/n) + (t-x_{i_0})(i_0-1)/n} where \eqn{i_0 = \max_{i=1,\ldots,m} \{x_i \le t\}}{i_0 = \max_{i=1,\ldots, m}{x_i \le t}}. } \usage{intECDF(s, x)} \arguments{ \item{s}{Vector of real numbers in \eqn{[x_1,x_m]} where \eqn{\bar{I}} should be evaluated at.} \item{x}{Vector \eqn{{\bold{x}} = (x_1, \ldots, x_m)}{x = (x_1, \ldots, x_m)} of original observations.} } \value{Vector of the same length as \eqn{\bold{s}}{s}, containing the values of \eqn{\bar I} at the elements of \eqn{\bold{s}}{s}.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \seealso{ This function together with \code{\link{intF}} can be used to check the characterization of the log-concave density estimator in terms of distribution functions, see Rufibach (2006) and Duembgen and Rufibach (2009). } \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} Rufibach K. (2006) \emph{Log-concave Density Estimation and Bump Hunting for i.i.d. Observations.} PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. \cr Available at \url{https://slsp-ube.primo.exlibrisgroup.com/permalink/41SLSP_UBE/17e6d97/alma99116730175505511}. } \examples{ # for an example see the function intF. } \keyword{htest} \keyword{nonparametric} logcondens/man/logconTwoSample.Rd0000644000176200001440000000547214130622661016575 0ustar liggesusers\name{logconTwoSample} \alias{logconTwoSample} \title{Compute p-values for two-sample test based on log-concave CDF estimates} \description{Compute \eqn{p}-values for a test for the null hypothesis of equal CDFs of two samples. The test statistic is reminiscient of Kolmogorv-Smirnov's, but instead of computing it for the empirical CDFs, this function computes it based on log-concave estimates for the CDFs.} \usage{logconTwoSample(x, y, which = c("MLE", "smooth"), M = 999, n.grid = 500, display = TRUE, seed0 = 1977)} \arguments{ \item{x}{First data sample.} \item{y}{Second data sample.} \item{which}{Indicate for which type of estimate the test statistic should be computed.} \item{M}{Number of permutations.} \item{n.grid}{Number of grid points in computation of maximal difference between smoothed log-concave CDFs. See \code{\link{maxDiffCDF}} for details.} \item{display}{If \code{TRUE} progress of computations is shown.} \item{seed0}{Set seed to reproduce results.} } \details{Given two i.i.d. samples \eqn{x_1, \ldots, x_{n_1}} and \eqn{y_1, \ldots, y_{n_2}} this function computes a permutation test \eqn{p}-value that provides evidence against the null hypothesis \deqn{H_0 : F_1 = F_2} where \eqn{F_1, F_2} are the CDFs of the samples, respectively. A test either based on the log-concave MLE or on its smoothed version (see Duembgen and Rufibach, 2009, Section 3) are provided. Note that computation of the smoothed version takes considerably more time. } \value{ \item{p.value}{A two dimensional vector containing the \eqn{p}-values.} \item{test.stat.orig}{The test statistics for the original samples.} \item{test.stats}{A \eqn{M \times 2} matrix containing the test statistics for all the permutations.} } \section{Warning}{Note that the algorithm that finds the maximal difference for the smoothed estimate is of approximative nature only. It may fail for very large sample sizes.} \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \examples{ \dontrun{ n1 <- 30 n2 <- 25 x <- rgamma(n1, 2, 1) y <- rgamma(n2, 2, 1) + 1 twosample <- logconTwoSample(x, y, which = c("MLE", "smooth")[1], M = 999) } } \keyword{htest} \keyword{nonparametric} logcondens/man/confIntBootLogConROC_t0.Rd0000644000176200001440000000623112615675116017760 0ustar liggesusers\name{confIntBootLogConROC_t0} \alias{confIntBootLogConROC_t0} \title{Function to compute a bootstrap confidence interval for the ROC curve at a given t, based on the log-concave ROC curve} \description{This function computes a bootstrap confidence interval for the ROC curve at a given value false negative fraction (1 - specificity) \eqn{t}. The ROC curve estimate is based on log-concave densities, as discussed in Rufibach (2011).} \usage{confIntBootLogConROC_t0(controls, cases, grid = c(0.2, 0.8), conf.level = 0.95, M = 1000, smooth = TRUE, output = TRUE)} \arguments{ \item{cases}{Values of the continuous variable for the cases.} \item{controls}{Values of the continuous variable for the controls.} \item{grid}{Values of 1 - specificity where confidence intervals should be computed at (may be a vector).} \item{conf.level}{Confidence level of confidence interval.} \item{M}{Number of bootstrap replicates.} \item{smooth}{\code{Logical}. Compute confidence interval also for ROC curve estimate based on smoothed log-concave densities.} \item{output}{\code{Logical}. Show progress of computations?} } \value{A list containing the following elements: \item{qs}{\code{data.frame} with the columns \code{t} (false positive fractions where confidence interval is computed at) and the confidence intervals for the ROC curve at \code{grid}, based on the log-concave density estimate.} \item{boot.mat}{Bootstrap samples for the ROC curve based on the log-concave density estimate.} \item{qs.smooth}{If \code{smooth = TRUE}, same as \code{qs} but for the ROC curve based on the smooth log-concave density estimate.} \item{boot.mat.smooth}{If \code{smooth = TRUE}, bootstrap samples for the ROC curve based on the smoothed log-concave density estimate.} } \references{The reference for computation of these bootstrap confidence intervals is: Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. \emph{Int. J. Biostat.}, \bold{8}(1), 1--29. The bootstrap competitor based on the empirical ROC curve is described in: Zhou, X.H. and Qin, G. (2005). Improved confidence intervals for the sensitivity at a fixed level of specificity of a continuous-scale diagnostic test. \emph{Statist. Med.}, \bold{24}, 465--477. } \author{Kaspar Rufibach (maintainer) \cr \email{kaspar.rufibach@gmail.com} \cr \url{http://www.kasparrufibach.ch}.} \note{The confidence intervals are only valid if observations are \emph{independent}, i.e. eacht patient only contributes one measurement, e.g.} \seealso{The ROC curve based on log-concave density estimates can be computed using \code{\link{logConROC}}. In the example below we analyze the \code{\link{pancreas}} data.} \examples{ \dontrun{ ## ROC curve for pancreas data data(pancreas) status <- factor(pancreas[, "status"], levels = 0:1, labels = c("healthy", "diseased")) var <- log(pancreas[, "ca199"]) cases <- var[status == "diseased"] controls <- var[status == "healthy"] ## compute confidence intervals res <- confIntBootLogConROC_t0(controls, cases, grid = c(0.2, 0.8), conf.level = 0.95, M = 1000, smooth = TRUE, output = TRUE) res } } \keyword{htest} logcondens/man/Local_LL_all.Rd0000644000176200001440000000301614130622661015721 0ustar liggesusers\name{Local_LL_all} \alias{Local_LL_all} \title{Log-likelihood, New Candidate and Directional Derivative for L} \description{ Computes the value of the log-likelihood function \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{x_1}^{x_m} \exp(\phi(t)) dt,}{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - int_{x_1}^{x_m} exp(\phi(t)) dt,} a new candidate for \eqn{\phi} via the Newton method as well as the directional derivative of \eqn{{\bold{\phi}} \to L({\bold{\phi}})}{\phi \to L(\phi)} into that direction. } \usage{Local_LL_all(x, w, phi)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{w}{Optional vector of nonnegative weights corresponding to \eqn{{\bold{x}_m}}{x_m}.} \item{phi}{Some vector \eqn{{\bold{\phi}}}{\phi} of the same length as \eqn{{\bold{x}}}{x} and \eqn{{\bold{w}}}{w}.} } \value{ \item{ll}{Value \eqn{L(\phi)} of the log-likelihood function at \eqn{\phi.}} \item{phi_new}{New candidate for \eqn{\phi} via the Newton-method, using the complete Hessian matrix.} \item{dirderiv}{Directional derivative of \eqn{\phi \to L(\phi)} into the direction \eqn{\phi_{new}.}} } \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/Q00.Rd0000644000176200001440000000415614130622712014013 0ustar liggesusers\name{Q00} \alias{Q00} \title{Numerical Routine Q} \description{This function is used in the computation of \eqn{\widehat f^*}{\hat f^*} and \eqn{\widehat F^*}{\hat F^*}.} \usage{ Q00(x, a, u, v, gamma, QFhat = FALSE)} \arguments{ \item{x}{Number at which to compute \eqn{q} and/or \eqn{Q}.} \item{a}{Vector of length \eqn{m} with real entries.} \item{u}{Vector of length \eqn{m} with real entries.} \item{v}{Vector of length \eqn{m} with real entries.} \item{gamma}{Real number. Standard deviation to be used.} \item{QFhat}{Logical. Should \eqn{Q} be computed?} } \value{The vector(s) \eqn{q} and/or \eqn{Q}.} \note{Taylor approximation is used if \eqn{a} is small. In addition, as described in Duembgen and Rufibach (2011) at the end of Appendix C, in extreme situations, e.g. when data sets contain extreme spacings, numerical problems may occur in the computation of the function \eqn{q_\gamma} (eq. (7) in Duembgen and Rufibach, 2011). For it may happen that the exponent is rather large while the difference of Gaussian CDFs is very small. To moderate these problems, we are using the following bounds: \deqn{\exp(- m^2/2) \bigl( \Phi(\delta) - \Phi(-\delta) \bigr) \ \le \ \Phi(b) - \Phi(a) \ \le \ \exp(- m^2/2) \cosh(m\delta) \bigl( \Phi(\delta) - \Phi(-\delta) \bigr)}{exp(- m ^ 2 / 2) (\Phi(\delta) - \Phi(-\delta)) \le \Phi(b) - \Phi(a) \le exp(- m^2/2) cosh(m\delta) ( \Phi(\delta) - \Phi(-\delta)} for arbitrary numbers \eqn{a < b} and \eqn{m := (a + b) / 2}, \eqn{\delta := (b - a) / 2}. However, the function \code{Q00} is not intended to be invoked by the end user. } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \references{ Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \url{https://www.jstatsoft.org/v39/i06} } \keyword{htest} \keyword{nonparametric} logcondens/man/summary.dlc.Rd0000644000176200001440000000353214130622661015711 0ustar liggesusers\name{summary.dlc} \Rdversion{1.1} \alias{summary.dlc} \title{Summarizing log-concave density estimation} \description{\code{summary} method for class \code{"dlc"}.} \usage{\method{summary}{dlc}(object, \dots)} \arguments{ \item{object}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens}.} \item{\dots}{Further arguments.} } \details{See \code{\link{activeSetLogCon}} and \code{\link{evaluateLogConDens}} for details on the computations.} \value{The function \code{summary.dlc} returns a list of summary statistics of the estimated log-concave density as well as of its smoothed version (depending on the value of \code{smoothed} when calling \code{\link{logConDens}}).} \section{Warning}{Note that the numbering of knots in the output relies on the vector of \emph{unique} observations.} \references{ Duembgen, L, Huesler, A. and Rufibach, K. (2010). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}. Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \examples{ ## See help file of function "logConDens". } \keyword{htest} \keyword{nonparametric} logcondens/man/robust.Rd0000644000176200001440000000344614130622661014775 0ustar liggesusers\name{robust} \alias{robust} \title{Robustification and Hermite Interpolation for ICMA} \description{Performs robustification and Hermite interpolation in the iterative convex minorant algorithm as described in Rufibach (2006, 2007).} \usage{robust(x, w, eta, etanew, grad)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{w}{Optional vector of nonnegative weights corresponding to \eqn{{\bold{x}_m}}{x_m}.} \item{eta}{Current candidate vector.} \item{etanew}{New candidate vector.} \item{grad}{Gradient of L at current candidate vector \eqn{\eta.}} } \value{Returns a (possibly) new vector \eqn{\eta} on the segment \deqn{(1 - t_0) \eta + t_0 \eta_{new} } such that the log-likelihood of this new \eqn{\eta} is strictly greater than that of the initial \eqn{\eta} and \eqn{t_0} is chosen according to the Hermite interpolation procedure described in Rufibach (2006, 2007).} \references{ Rufibach K. (2006) \emph{Log-concave Density Estimation and Bump Hunting for i.i.d. Observations.} PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. \cr Available at \url{https://slsp-ube.primo.exlibrisgroup.com/permalink/41SLSP_UBE/17e6d97/alma99116730175505511}. Rufibach, K. (2007) Computing maximum likelihood estimators of a log-concave density function. \emph{J. Stat. Comput. Simul.} \bold{77}, 561--574. } \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/Lhat_eta.Rd0000644000176200001440000000263414130622661015176 0ustar liggesusers\name{Lhat_eta} \alias{Lhat_eta} \title{Value of the Log-Likelihood Function L, where Input is in Eta-Parametrization} \description{ Gives the value of \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{x_1}^{x_m} \exp(\phi(t)) dt}{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - int_{x_1}^{x_m} exp(\phi(t)) dt} where \eqn{\phi} is parametrized via \deqn{{\bold{\eta}}({\bold{\phi}}) = \Bigl(\phi_1, \Bigl(\eta_1 + \sum_{j=2}^i (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m\Bigr).}{\eta(\phi) = (\phi_1, (\eta_1 + \sum_{j=2}^i (x_i-x_{i-1})\eta_i)_{i=2}^m).} } \usage{Lhat_eta(x, w, eta)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{w}{Optional vector of nonnegative weights corresponding to \eqn{{\bold{x}_m}}{x_m}.} \item{eta}{Some vector \eqn{{\bold{\eta}}}{\eta} of the same length as \eqn{{\bold{x}}}{x} and \eqn{{\bold{w}}}{w}.} } \value{Value \eqn{L({\bold{\phi}}) = L({\bold{\phi}}({\bold{\eta}}))}{L(\phi) = L(\phi(\eta))} of the log-likelihood function is returned.} \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/ROCx.Rd0000644000176200001440000000231212615675116014272 0ustar liggesusers\name{ROCx} \alias{ROCx} \title{Compute ROC curve at a given x based on log-concave estimates for the constituent distributions} \description{Computes the value of the ROC curve at \eqn{x} (which may be a vector) based on log-concave density estimates of the constituent distributions.} \usage{ROCx(x, res0, res1, smooth = FALSE)} \arguments{ \item{x}{Vector of numbers in \eqn{[0, 1]} where the ROC curve should be computed at.} \item{res0}{\code{dlc} object as a result of a call to \code{\link{logConDens}} for the data of the controls.} \item{res1}{\code{dlc} object as a result of a call to \code{\link{logConDens}} for the data of the cases.} \item{smooth}{Logical. If \code{TRUE} kernel smoothed log-concave estimate is used.} } \value{A real number or a vector of dimension the same as \eqn{x}, the value of the ROC curve at \code{x}.} \references{ Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. \emph{Int. J. Biostat.}, \bold{8}(1), 1--29. } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch}} \seealso{Used for the computation of AUC in \code{\link{logConROC}}.} \keyword{nonparametric} logcondens/man/Local_LL.Rd0000644000176200001440000000217014130622661015071 0ustar liggesusers\name{Local_LL} \alias{Local_LL} \title{Value of the Log-Likelihood Function L, where Input is in Phi-Parametrization} \description{ Gives the value of \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{x_1}^{x_m} \exp(\phi(t)) dt.}{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - int_{x_1}^{x_m} \exp(\phi(t)) dt.} } \usage{Local_LL(x, w, phi)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{w}{Optional vector of nonnegative weights corresponding to \eqn{{\bold{x}_m}}{x_m}.} \item{phi}{Some vector \eqn{{\bold{\phi}}}{\phi} of the same length as \eqn{{\bold{x}}}{x} and \eqn{{\bold{w}}}{w}.} } \value{Value \eqn{L=L({\bold{\phi}})}{L=L(\phi)} of the log-likelihood function is returned.} \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/logcondens-package.Rd0000644000176200001440000001730214130622661017177 0ustar liggesusers\name{logcon-package} \alias{logcon-package} \alias{logcon} \alias{log-concave} \alias{logcondens} \docType{package} \title{Estimate a Log-Concave Probability Density from iid Observations} \description{The main function of this package is \code{\link{logConDens}}: compute the maximum likelihood estimator (MLE) of a log-concave density from one-dimensional i.i.d. observations as well as the kernel smoothed version derived from it. A list of additional functions that can be used to compute quantities relevant in that context can be found below. Two algorithms are offered to compute the estimate: An active set (\code{\link{logConDens}}) and an iterative algorithm based on the pool-adjacent-violaters algorithm (\code{\link{icmaLogCon}}). The latter of these functions is only part of this package for historical reasons: it was the first method that was proposed to estimate a log-concave density, see Rufibach (2007). The more efficient way of computing the estimate is via an active set algorithm. The smoothed versions of the log-concave density and distribution function estimates discussed in Section 3 of Duembgen and Rufibach (2009) are available in the function \code{\link{evaluateLogConDens}}. To compute a log-concave density, CDF, and survival function from interval- or right-censored observations use the package \pkg{logconcens}. A log-concave probability mass function can be computed using the package \pkg{logcondiscr}, see Balabdaoui et al (2012) for details. For the computation of a log-concave estimate in any dimension the package \pkg{LogConDEAD} can be used.} \details{ \tabular{ll}{ Package: \tab logcondens\cr Type: \tab Package\cr Version: \tab 2.1.6 \cr Date: \tab 2021-10-10 \cr License: \tab GPL (>=2) \cr } The following additional functions and datasets are provided in the package: \code{\link{evaluateLogConDens}} Computes the value of the estimated log-density, density, and distribution function of the MLE and the smoothed estimator at a given \code{x0}. \code{\link{quantilesLogConDens}} Computes quantiles of the estimated distribution function at a given \code{x0}. \code{\link{intECDF}} Compute the integrated empirical distribution function of the observations at a given \code{x0}. \code{\link{intF}} Compute the integral of the distribution function corresponding to the log-concave density estimator at a given \code{x0}. \code{\link{logconTwoSample}} Compute a permutation test for the difference between two distribution functions. \code{\link{logConROC}} Compute ROC curve based on log-concave density estimates within cases and controls. \code{\link{confIntBootLogConROC_t0}} Compute bootstrap confidence intervals at given false positive fractions (= 1 - specificity) for the ROC curve based on log-concave density estimates. \code{\link{logConCI}} Compute a confidence interval for the value of the true density at a fixed point, based on the theory developed in Balabdaoui et al (2009). This function was contributed by Mahdis Azadbakhsh and Hanna Jankowski, see Azadbakhsh et al (2012). \code{\link{isoMean}} Compute the weighted least squares regression under a monotonicity constraint (the Grenander estimator). This function is used as part of \code{\link{icmaLogCon}} but is also of independent interest. The following datasets have been used in several publications to illustrate log-concave density estimation and are therefore included in this package: \code{\link{reliability}} Dataset that contains the data analyzed in Duembgen and Rufibach (2009, Figure 2). See the help file for \code{\link{logConDens}} for the analysis of this data. \code{\link{brightstar}} Dataset that contains the data analyzed in Mizera and Koenker (2009, Section 5). The sample consists of measurements of radial and rotational velocities for the stars from the Bright Star Catalog, see Hoffleit and Warren (1991). \code{\link{pancreas}} Data from pancreatic cancer serum biomarker study, first published by Wieand et al (1989). Contains data on serum measurements for a cancer antigen (CA-125) and a carbohydrate antigen (CA19.9) both of which are measured on a continuous non-negative scale. The measurements were taken within a case-control study on 90 cases with pancreatic cancer and 51 controls who did not have cancer but pancreatitis. A \code{\link{print}} and \code{\link{summary}} for objects of class \code{dlc}, generated by \code{\link{logConDens}}, are available as well. } \author{ Kaspar Rufibach (maintainer), \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} Kaspar Rufibach gratefully acknowledges support by the Swiss National Science Foundation SNF, \url{http://www.snf.ch}. MatLab code with an implementation of the active set algorithm is available on request from Lutz Duembgen.} \references{ Azadbakhsh, M., Jankowski, H. and Gao, X. (2014). Computing confidence intervals for log-concave densities. \emph{Comput. Statist. Data Anal.}, \bold{75}, 248--264. Balabdaoui, F., Jankowski, H., Rufibach, K. and Pavlides, M. (2012). Asymptotics of the discrete log-concave maximum likelihood estimator and related applications. \emph{J. R. Stat. Soc. Ser. B Stat. Methodol.}, \bold{75(4)}, 769--790. Baladbaoui, F., Rufibach, K. and Wellner, J. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density. \emph{Ann. Statist.}, \bold{37(3)}, 1299--1331. Duembgen, L., Huesler, A. and Rufibach, K. (2010). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}. Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011). logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} Hoffleit, D., Warren, W.H. (1991). \emph{The Bright Star Catalog.} Yale University Observatory, New Heaven. Mizera, I., Koenker, R. (2010). Quasi-concave density estimation. \emph{Ann. Statist.}, \bold{38(5)}, 2998--3027. Rufibach K. (2006). \emph{Log-concave Density Estimation and Bump Hunting for i.i.d. Observations.} PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. \cr Available at \url{https://slsp-ube.primo.exlibrisgroup.com/permalink/41SLSP_UBE/17e6d97/alma99116730175505511}. Rufibach, K. (2007). Computing maximum likelihood estimators of a log-concave density function. \emph{J. Stat. Comput. Simul.} \bold{77}, 561--574. Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. \emph{Int. J. Biostat.}, \bold{8}(1), 1--29. } \examples{ ## estimate gamma density set.seed(1977) x <- rgamma(100, 2, 1) res <- logConDens(x, smoothed = FALSE, print = TRUE) summary(res) ## compare performance to ICMA res2 <- icmaLogCon(x, T1 = 2000, robustif = TRUE, print = TRUE) res$L res2$L ## plot resulting functions par(mfrow = c(2, 2), mar = c(3, 2, 1, 2)) plot(res, which = "density") plot(res, which = "log-density") plot(res, which = "CDF") plot(res$x, res$H, type = 'l'); rug(res$x) ## compute function values at an arbitrary point x0 <- (res$x[50] + res$x[51]) / 2 evaluateLogConDens(x0, res) ## compute 0.5 quantile of Fhat quantilesLogConDens(0.5, res) } \keyword{htest} \keyword{nonparametric} logcondens/man/maxDiffCDF.Rd0000644000176200001440000000677214130622661015357 0ustar liggesusers\name{maxDiffCDF} \alias{maxDiffCDF} \title{Compute maximal difference between CDFs corresponding to log-concave estimates} \description{Compute the maximal difference between two estimated log-concave distribution functions, either the MLEs or the smoothed versions. This function is used to set up a two-sample permutation test that assesses the null hypothesis of equality of distribution functions.} \usage{maxDiffCDF(res1, res2, which = c("MLE", "smooth"), n.grid = 500)} \arguments{ \item{res1}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens} for the first sample.} \item{res2}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens} for the second sample.} \item{which}{Indicate for which type of estimate the maximal difference should be computed.} \item{n.grid}{Number of grid points used to find zeros of \eqn{\hat f_{n_1}^* - \hat f_{n_2}^*} for the smooth estimate.} } \details{Given two i.i.d. samples \eqn{x_1, \ldots, x_{n_1}} and \eqn{y_1, \ldots, y_{n_2}} this function computes the maxima of \deqn{D_1(t) = \hat F_{n_1}(t) - \hat F_{n_2}(t)} and \deqn{D_2(t) = \hat F^*_{n_1}(t) - \hat F^*_{n_2}(t).} } \value{ \item{test.stat}{A two-dimensional vector containing the above maxima.} \item{location}{A two-dimensional vector where the maxima occur.} } \section{Warning}{Note that the algorithm that finds the maximal difference for the smoothed estimate is of approximative nature only. It may fail for very large sample sizes.} \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \examples{ n1 <- 100 n2 <- 120 x <- sort(rgamma(n1, 2, 1)) y <- sort(rgamma(n2, 2, 1)) res1 <- logConDens(x, smoothed = TRUE) res2 <- logConDens(y, smoothed = TRUE) d <- maxDiffCDF(res1, res2, n.grid = 200) ## log-concave estimate xs <- seq(min(res1$xs, res2$xs), max(res1$xs, res2$xs), length = 200) F1 <- matrix(NA, nrow = length(xs), ncol = 1); F2 <- F1 for (i in 1:length(xs)){ F1[i] <- evaluateLogConDens(xs[i], res1, which = 3)[, "CDF"] F2[i] <- evaluateLogConDens(xs[i], res2, which = 3)[, "CDF"] } par(mfrow = c(1, 2)) plot(xs, abs(F1 - F2), type = "l") abline(v = d$location[1], lty = 2, col = 3) abline(h = d$test.stat[1], lty = 2, col = 3) ## smooth estimate xs <- seq(min(res1$xs, res2$xs), max(res1$xs, res2$xs), length = 200) F1smooth <- matrix(NA, nrow = length(xs), ncol = 2); F2smooth <- F1smooth for (i in 1:length(xs)){ F1smooth[i, ] <- evaluateLogConDens(xs[i], res1, which = 4:5)[, c("smooth.density", "smooth.CDF")] F2smooth[i, ] <- evaluateLogConDens(xs[i], res2, which = 4:5)[, c("smooth.density", "smooth.CDF")] } plot(xs, abs(F1smooth[, 2] - F2smooth[, 2]), type = "l") abline(h = 0) abline(v = d$location[2], lty = 2, col = c(3, 4)) abline(h = d$test.stat[2], lty = 2, col = c(3, 4)) } \keyword{htest} \keyword{nonparametric} logcondens/man/MLE.Rd0000644000176200001440000000434314130622661014071 0ustar liggesusers\name{MLE} \alias{MLE} \title{Unconstrained piecewise linear MLE} \description{ Given a vector of observations \eqn{{\bold{x}} = (x_1, \ldots, x_m)}{x = (x_1, \ldots, x_m)} with pairwise distinct entries and a vector of weights \eqn{{\bold{w}}=(w_1, \ldots, w_m)}{w =(w_1, \ldots, w_m)} s.t. \eqn{\sum_{i=1}^m w_i = 1}, this function computes a function \eqn{\widehat \phi_{MLE}}{\hat \phi_{MLE}} (represented by the vector \eqn{(\widehat \phi_{MLE}(x_i))_{i=1}^m}{(\hat \phi_{MLE}(x_i))_{i=1}^m}) supported by \eqn{[x_1, x_m]} such that \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \sum_{j=1}^{m-1} (x_{j+1} - x_j) J(\phi_j, \phi_{j+1})} is maximal over all continuous, piecewise linear functions with knots in \eqn{\{x_1, \ldots, x_m\}}{{x_1, \ldots, x_m}.} } \usage{MLE(x, w = NA, phi_o = NA, prec = 1e-7, print = FALSE)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{w}{Optional vector of nonnegative weights corresponding to \eqn{{\bold{x}_m}}{x_m}.} \item{phi_o}{Optional starting vector.} \item{prec}{Threshold for the directional derivative during the Newton-Raphson procedure.} \item{print}{print = TRUE outputs log-likelihood in every loop, print = FALSE does not. Make sure to tell R to output (press CTRL+W).} } \value{ \item{phi}{Resulting column vector \eqn{(\widehat \phi_{MLE}(x_i))_{i=1}^m.}{(\hat \phi_{MLE}(x_i))_{i=1}^m.}} \item{L}{Value \eqn{L(\widehat \phi_{MLE})}{L(\hat \phi_{MLE})} of the log-likelihood at \eqn{\widehat \phi_{MLE}.}{\hat \phi_{MLE}.}} \item{Fhat}{Vector of the same length as \eqn{{\bold{x}}}{x} with entries \eqn{\widehat F_{MLE,1} = 0}{\hat F_{MLE,1} = 0} and \deqn{\widehat F_{MLE,k} = \sum_{j=1}^{k-1} (x_{j+1} - x_j) J(\phi_j, \phi_{j+1}) }{\hat F_{MLE,k} = \sum_{j=1}^{k-1} (x_{j+1} - x_j) J(\phi_j, \phi_{j+1})} for \eqn{k \ge 2.}{k >= 2.}} } \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/intF.Rd0000644000176200001440000000673414130622661014362 0ustar liggesusers\name{intF} \alias{intF} \title{Computes the Integral of the estimated CDF at Arbitrary Real Numbers in s} \description{Based on an object of class \code{dlc} as output by the function \code{\link{logConDens}}, this function gives values of \deqn{\widehat I(t) = \int_{x_1}^t \widehat{F}(r) d r}{\hat I(t) = \int_{x_1}^t \hat{F}(r) d r} at all numbers in \eqn{\bold{s}}{s}. Note that \eqn{t} (so all elements in \eqn{\bold{s}}{s}) must lie in \eqn{[x_1,x_m]}. The exact formula for \eqn{\widehat I(t)} is \deqn{\widehat I(t) = \Bigl(\sum_{i=1}^{i_0} \widehat{I}_i(x_{i+1})\Bigr)+\widehat{I}_{i_0}(t)}{\hat I(t) = (\sum_{i=1}^{i_0} \hat{I}_i(x_{i+1}))+\hat{I}_{i_0}(t)} where \eqn{i_0 = }min\eqn{\{m-1 \, , \ \{i \ : \ x_i \le t \}\}}{{m-1, {i : x_i \le t}}} and \deqn{I_j(x) = \int_{x_j}^x \widehat{F}(r) d r = (x-x_j)\widehat{F}(x_j)+\Delta x_{j+1}\Bigl(\frac{\Delta x_{j+1}}{\Delta \widehat\phi_{j+1}}J\Bigl(\widehat\phi_j,\widehat\phi_{j+1}, \frac{x-x_j}{\Delta x_{j+1}}\Bigr)-\frac{\widehat f(x_j)(x-x_j)}{\Delta \widehat \phi_{j+1}}\Bigr)}{I_j(x) = int_{x_j}^x \hat{F}(r) d r = (x-x_j)\hat{F}(x_j)+\Delta x_{j+1}((\Delta x_{j+1})/(\Delta \hat\phi_{j+1})J(\hat\phi_j, \hat \phi_{j+1}, (x-x_j)/(\Delta x_{j+1}))-(\hat f(x_j)(x-x_j))/(\Delta \hat \phi_{j+1}))} for \eqn{x \in [x_j, x_{j+1}], \ j = 1,\ldots, m-1}{x \in [x_j, x_{j+1}], j = 1, \ldots, m-1}, \eqn{\Delta v_{i+1} = v_{i+1} - v_i} for any vector \eqn{\bold{v}}{v} and the function \eqn{J} introduced in \code{\link{Jfunctions}}. } \usage{intF(s, res)} \arguments{ \item{s}{Vector of real numbers where the functions should be evaluated at.} \item{res}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens}.} } \value{Vector of the same length as \eqn{\bold{s}}, containing the values of \eqn{\widehat I} at the elements of \eqn{\bold{s}}{s}.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \seealso{ This function uses the output of \code{\link{activeSetLogCon}}. The function \code{\link{intECDF}} is similar, but based on the empirical distribution function. } \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} Rufibach K. (2006) \emph{Log-concave Density Estimation and Bump Hunting for i.i.d. Observations.} PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. \cr Available at \url{https://slsp-ube.primo.exlibrisgroup.com/permalink/41SLSP_UBE/17e6d97/alma99116730175505511}. } \examples{ ## estimate gamma density set.seed(1977) x <- rgamma(200, 2, 1) res <- logConDens(x, smoothed = FALSE, print = FALSE) ## compute and plot the process D(t) in Duembgen and Rufibach (2009) s <- seq(min(res$x), max(res$x), by = 10 ^ -3) D1 <- intF(s, res) D2 <- intECDF(s, res$xn) par(mfrow = c(2, 1)) plot(res$x, res$phi, type = 'l'); rug(res$x) plot(s, D1 - D2, type = 'l'); abline(h = 0, lty = 2) } \keyword{htest} \keyword{nonparametric} logcondens/man/brightstar.Rd0000644000176200001440000000266712615675116015645 0ustar liggesusers\name{brightstar} \alias{brightstar} \docType{data} \title{Bright star dataset used to illustrate log-concave density estimation} \description{Dataset that contains the data analyzed in Mizera and Koenker (2009, Section 5). The sample consists of measurements of radial and rotational velocities for the stars from the Bright Star Catalog, see Hoffleit and Warren (1991).} \usage{data(brightstar)} \format{ A data frame with 9092 rows on the following 2 variables. \describe{ \item{\code{nr}}{Location of measurements.} \item{\code{rad}}{Measurements of radial velocities.} \item{\code{rot}}{Measurements of rotational velocities.} } } \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Hoffleit, D., Warren, W.H. (1991). \emph{The Bright Star Catalog.} Yale University Observatory, New Heaven. Mizera, I., Koenker, R. (2010). Quasi-concave density estimation. \emph{Ann. Statist.}, \bold{38(5)}, 2998--3027. } \examples{ # ---- load rotational velocity data ---- data(brightstar) # ---- compute and plot log-concave estimate ---- # See also Figure 3 in Koenker & Mizera (2009) x0 <- sort(brightstar[, 3]) res <- logConDens(x0, print = FALSE, smoothed = FALSE) plot(res, which = "density") } \keyword{datasets} logcondens/man/evaluateLogConDens.Rd0000644000176200001440000000600014130622661017166 0ustar liggesusers\name{evaluateLogConDens} \alias{evaluateLogConDens} \title{Evaluates the Log-Density MLE and Smoothed Estimator at Arbitrary Real Numbers xs} \description{Based on a \code{"dlc"} object generated by \code{\link{logConDens}}, this function computes the values of \deqn{\widehat \phi_m(t)}{\hat \phi_m(t)} \deqn{\widehat f_m(t) = \exp(\widehat \phi_m(t))}{\hat f_m(t) = exp(\hat \phi_m(t))} \deqn{\widehat F_m(t) = \int_{x_1}^t \exp(\widehat \phi_m(x)) dx}{\hat F_m(t) = int_{x_1}^t exp(\hat \phi_m(x)) dx} \deqn{\widehat f_m^*(t) = \exp(\widehat \phi_m^*(t))}{\hat f_m^*(t) = exp(\hat \phi_m^*(t))} \deqn{\widehat F_m^*(t) = \int_{x_1}^t \exp(\widehat \phi_m^*(x)) dx}{\hat F_m^*(t) = int_{x_1}^t \exp(\hat \phi_m^*(x)) dx} at all real number \eqn{t} in \code{xs}. The exact formula for \eqn{\widehat F_m}{\hat F_m} and \eqn{t \in [x_j,x_{j+1}]} is \deqn{\widehat F_m(t) = \widehat F_m(x_j) + (x_{j+1}-x_j) J\Big(\widehat \phi_j, \widehat \phi_{j+1}, \frac{t-x_j}{x_{j+1}-x_j} \Big)}{\hat F_m(t) = \hat F_m(x_j) + (x_{j+1}-x_j) J(\hat \phi_j, \hat \phi_{j+1}, (t-x_j)/(x_{j+1}-x_j))} for the function \eqn{J} introduced in \code{\link{Jfunctions}}. Closed formulas can also be given for \eqn{\widehat f_m^*(t)}{\hat f_m^*(t)} and \eqn{\widehat F_m^*(t)}{\hat F_m^*(t)}. } \usage{evaluateLogConDens(xs, res, which = 1:5, gam = NULL, print = FALSE)} \arguments{ \item{xs}{Vector of real numbers where the functions should be evaluated at.} \item{res}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens}.} \item{which}{A (sub-)vector of \code{1:5} specifying which of the above quantities should be computed.} \item{gam}{Only necessary if \code{smoothed = TRUE}. The standard deviation of the normal kernel. If equal to \code{NULL}, \code{gam} is chosen such that the variances of the original sample \eqn{x_1, \ldots, x_n} and \eqn{\widehat f_n^*}{\hat f_n^*} coincide. See \code{\link{logConDens}} for details.} \item{print}{Progress in computation of smooth estimates is shown.} } \value{Matrix with rows \eqn{(x_{0, i}, \widehat \phi_m(x_{0, i}), \widehat f_m(x_{0, i}), \widehat F_m(x_{0, i}), \widehat f_m^*(x_{0, i}), \widehat F_m^*(x_{0, i}))}{(x_{0, i}, \hat \phi_m(x_{0, i}), \hat f_m(x_{0, i}), \hat F_m(x_{0, i}), \hat f_m^*(x_{0, i}), \hat F_m^*(x_{0, i}))} where \eqn{x_{0,i}} is the \eqn{i}-th entry of \code{xs}.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \examples{ ## estimate gamma density set.seed(1977) x <- rgamma(200, 2, 1) res <- logConDens(x, smoothed = TRUE, print = FALSE) ## compute function values at an arbitrary point xs <- (res$x[100] + res$x[101]) / 2 evaluateLogConDens(xs, res) ## only compute function values for non-smooth estimates evaluateLogConDens(xs, res, which = 1:3) } \keyword{htest} \keyword{nonparametric} logcondens/man/activeSetRoutines.Rd0000644000176200001440000000323514130622661017133 0ustar liggesusers\name{activeSetRoutines} \alias{activeSetRoutines} \alias{LocalCoarsen} \alias{LocalConvexity} \alias{LocalExtend} \alias{LocalF} \alias{LocalNormalize} \alias{LocalMLE} \alias{LocalVariance} \title{Auxiliary Numerical Routines for the Function activeSetLogCon} \description{Functions that are used by activeSetLogCon.} \usage{ LocalCoarsen(x, w, IsKnot) LocalConvexity(x, phi) LocalExtend(x, IsKnot, x2, phi2) LocalF(x, phi) LocalNormalize(x, phi) LocalMLE(x, w, IsKnot, phi_o, prec) LocalVariance(x, w = NULL, phi) } \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{w}{Optional vector of nonnegative weights corresponding to \eqn{{\bold{x}}}{x}.} \item{IsKnot}{Vector with entries IsKnot\eqn{_i = 1\{\phi} has a kink at \eqn{x_i\}.}} \item{phi}{Vector with entries \eqn{\phi(x_i).}} \item{x2}{Vector of same type as \eqn{{\bold{x}}}{x}.} \item{phi2}{Vector of same type as \eqn{{\bold{\phi}}}{\phi}.} \item{phi_o}{Optional starting vector.} \item{prec}{Threshold for the directional derivative during Newton-Raphson procedure.} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \seealso{All the above functions are used by \code{\link{activeSetLogCon}} to estimate a log-concave probability density. Log concave density estimation via an iterative convex minorant algorithm can be performed using \code{\link{icmaLogCon}}. } \keyword{htest} \keyword{nonparametric} logcondens/man/icmaLogCon.Rd0000644000176200001440000001100514130622661015460 0ustar liggesusers\name{icmaLogCon} \alias{icmaLogCon} \title{Computes a Log-Concave Probability Density Estimate via an Iterative Convex Minorant Algorithm} \description{Given a vector of observations \eqn{{\bold{x}_n} = (x_1, \ldots, x_n)}{x_n = (x_1, \ldots, x_n)} with not necessarily equal entries, \code{\link{activeSetLogCon}} first computes vectors \eqn{{\bold{x}_m} = (x_1, \ldots, x_m)}{x_m = (x_1, \ldots, x_m)} and \eqn{{\bold{w}} = (w_1, \ldots, w_m)}{w = (w_1, \ldots, w_m)} where \eqn{w_i} is the weight of each \eqn{x_i} s.t. \eqn{\sum_{i=1}^m w_i = 1}. Then, \code{\link{activeSetLogCon}} computes a concave, piecewise linear function \eqn{\widehat \phi_m} on \eqn{[x_1, x_m]} with knots only in \eqn{\{x_1, \ldots, x_m\}}{{x_1, \ldots, x_m}} such that \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{-\infty}^\infty \exp(\phi(t)) dt}{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{-\infty}^\infty exp(\phi(t)) dt} is maximal. In order to be able to apply the pool - adjacent - violaters algorithm, computations are performed in the parametrization \deqn{{\bold{\eta}}({\bold{\phi}}) = \Bigl(\phi_1, \Bigl(\eta_1 + \sum_{j=2}^i (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m \Bigr).}{\eta(\phi) = (\phi_1, (\eta_1 + \sum_{j=2}^i (x_i-x_{i-1})\eta_i)_{i=2}^m ).} To find the maximum of \eqn{L}, a variant of the iterative convex minorant using the pool - adjacent - violaters algorithm is used.} \usage{icmaLogCon(x, xgrid = NULL, eps = 10^-8, T1 = 2000, robustif = TRUE, print = FALSE)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, not necessarily equal.} \item{xgrid}{Governs the generation of weights for observations. See \code{\link{preProcess}} for details.} \item{eps}{An arbitrary real number, typically small. Iterations are halted if the directional derivative of \eqn{{\bold{\eta}} \to L({\bold{\eta}})} in the direction of the new candidate is \eqn{\le \varepsilon}.} \item{T1}{Maximal number of iterations to perform.} \item{robustif}{\code{robustif = TRUE} performs the robustification and Hermite interpolation procedure detailed in Rufibach (2006, 2007), \code{robustif = FALSE} does not. In the latter case, convergence of the algorithm is no longer guaranteed.} \item{print}{\code{print = TRUE} outputs log-likelihood in every loop, \code{print = FALSE} does not. Make sure to tell \code{R} to output (press CTRL+W).} } \value{ \item{x}{Vector of observations \eqn{x_1, \ldots, x_m} that was used to estimate the density. } \item{w}{The vector of weights that had been used. Depends on the chosen setting for \code{xgrid}.} \item{f}{Vector with entries \eqn{\widehat f_m(x_i).}} \item{xn}{Vector with initial observations \eqn{x_1, \ldots, x_n}.} \item{Loglik}{The value \eqn{L(\widehat \phi_m)} of the log-likelihood-function \eqn{L} at the maximum \eqn{\widehat \phi_m.}} \item{Iterations}{Number of iterations performed.} \item{sig}{The standard deviation of the initial sample \eqn{x_1, \ldots, x_n}.} } \references{ Rufibach K. (2006) \emph{Log-concave Density Estimation and Bump Hunting for i.i.d. Observations.} PhD Thesis, University of Bern, Switzerland and Georg-August University of Goettingen, Germany, 2006. \cr Available at \url{https://slsp-ube.primo.exlibrisgroup.com/permalink/41SLSP_UBE/17e6d97/alma99116730175505511}. Rufibach, K. (2007) Computing maximum likelihood estimators of a log-concave density function. \emph{J. Stat. Comput. Simul.} \bold{77}, 561--574. } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \seealso{ \code{\link{icmaLogCon}} can be used to estimate a log-concave density. However, to generate an object of class \code{dlc} that allows application of \code{\link{summary}} and \code{\link{plot}} one has to use \code{\link{logConDens}}. The following functions are used by \code{\link{icmaLogCon}}: \code{\link{phieta}}, \code{\link{etaphi}}, \code{\link{Lhat_eta}}, \code{\link{quadDeriv}}, \code{\link{robust}}, \code{\link{isoMean}}. } \examples{ set.seed(1977) x <- rgamma(200, 2, 1) \dontrun{ res <- icmaLogCon(x, T1 = 2000, robustif = TRUE, print = TRUE) ## plot resulting functions par(mfrow = c(2, 1), mar = c(3, 2, 1, 2)) plot(x, exp(res$phi), type = 'l'); rug(x) plot(x, res$phi, type = 'l'); rug(x) } } \keyword{htest} \keyword{nonparametric} logcondens/man/quadDeriv.Rd0000644000176200001440000000327414130622661015402 0ustar liggesusers\name{quadDeriv} \alias{quadDeriv} \title{Gradient and Diagonal of Hesse Matrix of Quadratic Approximation to Log-Likelihood Function L} \description{ Computes gradient and diagonal of the Hesse matrix w.r.t. to \eqn{\eta} of a quadratic approximation to the reparametrized original log-likelihood function \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{-\infty}^{\infty} \exp(\phi(t)) dt. }{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - int_{-\infty}^{\infty} exp(\phi(t)) dt. } where \eqn{L} is parametrized via \deqn{{\bold{\eta}}({\bold{\phi}}) = \Bigl(\phi_1, \Bigl(\eta_1+ \sum_{j=2}^i (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m\Bigr).}{\eta(\phi) = (\phi_1, (\eta_1 + \sum_{j=2}^i (x_i-x_{i-1}) \eta_i)_{i=2}^m).} \eqn{{\bold{\phi}}}{\phi}: vector \eqn{(\phi(x_i))_{i=1}^m} representing concave, piecewise linear function \eqn{\phi},\cr \eqn{{\bold{\eta}}}{\eta}: vector representing successive slopes of \eqn{\phi.} } \usage{quadDeriv(dx, w, eta)} \arguments{ \item{dx}{Vector \eqn{(0, x_i-x_{i-1})_{i=2}^m.}} \item{w}{Vector of weights as in \code{\link{activeSetLogCon}}.} \item{eta}{Vector \eqn{{\bold{\eta}}.}{\eta.}} } \value{\eqn{m \times 2} matrix. First column contains gradient and second column diagonal of Hesse matrix.} \seealso{\code{\link{quadDeriv}} is used by the function \code{\link{icmaLogCon}}.} \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/plot.dlc.Rd0000644000176200001440000000511614130622661015172 0ustar liggesusers\name{plot.dlc} \alias{plot.dlc} \title{Standard plots for a dlc object} \description{\code{plot} method for class \code{"dlc"}. Three plots (selectable by \code{which}) are currently available: a plot of the estimated density, the estimated log-density, or the distribution function corresponding to the estimated log-concave density. By default, a plot of the density estimate is provided. If \code{smoothed = TRUE}, the smoothed version of the log-concave density estimate (saved in \code{x}) is added to the density and log-density plot. For the CDF, the smoothed version is not contained by default in a \code{dlc} object and needs to be computed when asked to be plotted.} \usage{\method{plot}{dlc}(x, which = c("density", "log-density", "CDF"), add.title = TRUE, legend.pos = "topright", \dots)} \arguments{ \item{x}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens}.} \item{which}{One of \code{"density"}, \code{"log-density"}, or \code{"CDF"}.} \item{add.title}{Logical, if \code{TRUE} adds a standard title to the plot.} \item{legend.pos}{Placement of the legend. One of \code{"bottomright"}, \code{"bottom"}, \code{"bottomleft"}, \code{"left"}, \code{"topleft"}, \code{"top"}, \code{"topright"}, \code{"right"}, \code{"center"}; or \code{"none"} for not displaying a legend. See \code{\link{legend}} for details.} \item{\dots}{Further arguments.} } \details{See \code{\link{activeSetLogCon}} and \code{\link{evaluateLogConDens}} for details on the computations.} \value{Chosen plot is generated.} \references{ Duembgen, L, Huesler, A. and Rufibach, K. (2010). Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}. Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \examples{ ## See help file of function "logConDens". } \keyword{htest} \keyword{nonparametric} logcondens/man/isoMean.Rd0000644000176200001440000000273714130622661015054 0ustar liggesusers\name{isoMean} \alias{isoMean} \title{Pool-Adjacent Violaters Algorithm: Least Square Fit under Monotonicity Constraint} \description{ Fits a vector \eqn{\widehat {\bold{g}}}{\hat g} with nondecreasing components to the data vector \eqn{{\bold{y}}}{y} such that \deqn{\sum_{i=1}^n (y_i - \widehat g_i)^2 }{\sum_{i=1}^n (y_i - \hat g_i)^2 } is minimal (pool - adjacent - violators algorithm). In case a weight vector with positive entries (and the same size as \eqn{{\bold{y}}}{y}) is provided, the function produces an isotonic vector minimizing \deqn{\sum_{i=1}^n w_i(y_i - \widehat g_i)^2 .}{\sum_{i=1}^n w_i(y_i - \hat g_i)^2.} } \usage{isoMean(y, w)} \arguments{ \item{y}{Vector \eqn{(y_1, \ldots, y_n)} of data points.} \item{w}{Arbitrary vector \eqn{(w_1, \ldots, w_n)} of weights.} } \value{Returns vector \eqn{\widehat {\bold{g}}}{\widehat g}.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \examples{ ## simple regression model n <- 50 x <- sort(runif(n, 0, 1)) y <- x ^ 2 + rnorm(n, 0, 0.2) s <- seq(0, 1, by = 0.01) plot(s, s ^ 2, col = 2, type = 'l', xlim = range(c(0, 1, x)), ylim = range(c(0, 1 , y))); rug(x) ## plot pava result lines(x, isoMean(y, rep(1 / n, n)), type = 's') } \keyword{ htest } \keyword{ nonparametric } logcondens/man/quantilesLogConDens.Rd0000644000176200001440000000403414130622661017372 0ustar liggesusers\name{quantilesLogConDens} \alias{quantilesLogConDens} \title{Function to compute Quantiles of Fhat} \description{Function to compute \eqn{p_0}-quantile of \deqn{\widehat F_m(t) = \int_{x_1}^t \widehat f_m(t) dt,}{\hat F_m(t) = \int_{x_1}^t \hat f_m(t) dt,} where \eqn{\widehat f_m} is the log-concave density estimator, typically computed via \code{\link{logConDens}} and \eqn{p_0} runs through the vector \code{ps}. The formula to compute a quantile at \eqn{u \in [\widehat F_m(x_j), \widehat F_m(x_{j+1})]}{u \in [\hat F_m(x_j), \hat F_m(x_{j+1})]} for \eqn{j = 1, \ldots, n-1} is: \deqn{\widehat F_m^{-1}(u) = x_j + (x_{j+1}-x_j) G^{-1}_{(x_{j+1}-x_j)(\widehat \phi_{j+1}-\widehat \phi_j)} \Big( \frac{u - \widehat F_m(x_j)}{ \widehat F_m(x_{j+1}) - \widehat F_m(x_j)}\Big),}{\hat F_m^{-1}(u) = x_j + (x_{j+1}-x_j) G^{-1}_{(x_{j+1}-x_j)(\hat \phi_{j+1}-\hat \phi_j)} ((u - \hat F_m(x_j))/(\hat F_m(x_{j+1}) - \hat F_m(x_j))),} where \eqn{G^{-1}_\theta} is described in \code{\link{qloglin}}.} \usage{quantilesLogConDens(ps, res)} \arguments{ \item{ps}{Vector of real numbers where quantiles should be computed.} \item{res}{An object of class \code{"dlc"}, usually a result of a call to \code{logConDens}.} } \value{Returns a data.frame with row \eqn{(p_{0, i}, q_{0, i})} where \eqn{q_{0, i} = \inf_{x}\{\widehat F_m(x) \ge p_{0, i}\}}{q_{0, i} = inf_{x}\{\hat F_m(x) \ge p_{0, i}\}} and \eqn{p_{0, i}} runs through \code{ps}.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \examples{ ## estimate gamma density set.seed(1977) x <- rgamma(200, 2, 1) res <- logConDens(x, smoothed = FALSE, print = FALSE) ## compute 0.95 quantile of Fhat q <- quantilesLogConDens(0.95, res)[, "quantile"] plot(res, which = "CDF", legend.pos = "none") abline(h = 0.95, lty = 3); abline(v = q, lty = 3) } \keyword{htest} \keyword{nonparametric} logcondens/man/activeSetLogCon.Rd0000644000176200001440000001256614130622661016513 0ustar liggesusers\name{activeSetLogCon} \alias{activeSetLogCon} \alias{activeSet} \title{Computes a Log-Concave Probability Density Estimate via an Active Set Algorithm} \description{Given a vector of observations \eqn{{\bold{x}_n} = (x_1, \ldots, x_n)}{x_n = (x_1, \ldots, x_n)} with not necessarily equal entries, \code{\link{activeSetLogCon}} first computes vectors \eqn{{\bold{x}_m} = (x_1, \ldots, x_m)}{x_m = (x_1, \ldots, x_m)} and \eqn{{\bold{w}} = (w_1, \ldots, w_m)}{w = (w_1, \ldots, w_m)} where \eqn{w_i} is the weight of each \eqn{x_i} s.t. \eqn{\sum_{i=1}^m w_i = 1}. Then, \code{\link{activeSetLogCon}} computes a concave, piecewise linear function \eqn{\widehat \phi_m}{\widehat \phi_m} on \eqn{[x_1, x_m]} with knots only in \eqn{\{x_1, \ldots, x_m\}}{{x_1, \ldots, x_m}} such that \deqn{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - \int_{-\infty}^\infty \exp(\phi(t)) dt}{L(\phi) = \sum_{i=1}^m w_i \phi(x_i) - int_{-\infty}^\infty exp(\phi(t)) dt} is maximal. To accomplish this, an active set algorithm is used.} \usage{activeSetLogCon(x, xgrid = NULL, print = FALSE, w = NA)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, not necessarily unique.} \item{xgrid}{Governs the generation of weights for observations. See \code{\link{preProcess}} for details.} \item{print}{\code{print = TRUE} outputs the log-likelihood in every loop, \code{print = FALSE} does not. Make sure to tell \code{R} to output (press CTRL+W).} \item{w}{Optional vector of weights. If weights are provided, i.e. if \code{w != NA}, then \code{xgrid} is ignored.} } \value{ \item{xn}{Vector with initial observations \eqn{x_1, \ldots, x_n}.} \item{x}{Vector of observations \eqn{x_1, \ldots, x_m} that was used to estimate the density. } \item{w}{The vector of weights that had been used. Depends on the chosen setting for \code{xgrid}.} \item{phi}{Vector with entries \eqn{\widehat \phi_m(x_i)}.} \item{IsKnot}{Vector with entries IsKnot\eqn{_i = 1\{\widehat \phi_m} has a kink at \eqn{x_i\}}.} \item{L}{The value \eqn{L(\widehat {\bold{\phi}}_m)}{L(\phi_m)} of the log-likelihood-function \eqn{L} at the maximum \eqn{\widehat {\bold{\phi}}_m}{\widehat \phi_m}.} \item{Fhat}{A vector \eqn{(\widehat F_{m,i})_{i=1}^m} of the same size as \eqn{{\bold{x}}}{x} with entries \deqn{\widehat F_{m,i} = \int_{x_1}^{x_i} \exp(\widehat \phi_m(t)) dt.}} \item{H}{Vector \eqn{(H_1, \ldots, H_m)'} where \eqn{H_i} is the derivative of \deqn{t \to L(\phi + t\Delta_i)} at zero and \eqn{\Delta_i(x) = \min(x - x_i, 0).}} \item{n}{Number of initial observations.} \item{m}{Number of unique observations.} \item{knots}{Observations that correspond to the knots.} \item{mode}{Mode of the estimated density \eqn{\hat f_m}.} \item{sig}{The standard deviation of the initial sample \eqn{x_1, \ldots, x_n}.} } \references{ Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}. Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \seealso{\code{\link{activeSetLogCon}} can be used to estimate a log-concave density. However, to generate an object of class \code{dlc} that allows application of \code{\link{summary}} and \code{\link{plot}} we recommend to use \code{\link{logConDens}}. The following functions are used by \code{\link{activeSetLogCon}}: \code{\link{J00}}, \code{\link{J10}}, \code{\link{J11}}, \code{\link{J20}}, \code{\link{Local_LL}}, \code{\link{Local_LL_all}}, \code{\link{LocalCoarsen}}, \code{\link{LocalConvexity}}, \code{\link{LocalExtend}}, \code{\link{LocalF}}, \code{\link{LocalMLE}}, \code{\link{LocalNormalize}}, \code{\link{MLE}} Log concave density estimation via an iterative convex minorant algorithm can be performed using \code{\link{icmaLogCon}}. } \examples{ ## estimate gamma density set.seed(1977) n <- 200 x <- rgamma(n, 2, 1) res <- activeSetLogCon(x, w = rep(1 / n, n), print = FALSE) ## plot resulting functions par(mfrow = c(2, 2), mar = c(3, 2, 1, 2)) plot(res$x, exp(res$phi), type = 'l'); rug(x) plot(res$x, res$phi, type = 'l'); rug(x) plot(res$x, res$Fhat, type = 'l'); rug(x) plot(res$x, res$H, type = 'l'); rug(x) ## compute and plot function values at an arbitrary point x0 <- (res$x[100] + res$x[101]) / 2 Fx0 <- evaluateLogConDens(x0, res, which = 3)[, "CDF"] plot(res$x, res$Fhat, type = 'l'); rug(res$x) abline(v = x0, lty = 3); abline(h = Fx0, lty = 3) ## compute and plot 0.9-quantile of Fhat q <- quantilesLogConDens(0.9, res)[2] plot(res$x, res$Fhat, type = 'l'); rug(res$x) abline(h = 0.9, lty = 3); abline(v = q, lty = 3) } \keyword{htest} \keyword{nonparametric} logcondens/man/preProcess.Rd0000644000176200001440000000267414130622661015606 0ustar liggesusers\name{preProcess} \alias{preProcess} \title{Compute a weighted sample from initial observations} \description{Generates weights from initial sample.} \usage{preProcess(x, xgrid = NULL)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, not necessarily unique.} \item{xgrid}{Parameter that governs the generation of weights: If \code{xgrid = NULL} a new sample of unique observations is generated with corresponding vector of weights. If \code{xgrid} is a positive number, observations are binned in a grid with grid length \code{xgrid}. Finally, an entire vector specifying a user-defined grid can be supplied.} } \value{ \item{x}{Vector of unique and sorted observations deduced from the input \code{x} according to the specification given by \code{xgrid}.} \item{w}{Vector of corresponding weights, normalized to sum to one.} \item{sig}{Standard deviation of the inputed observations. This quantity is needed when computing the smoothed log-concave density estimator via \code{\link{evaluateLogConDens}}.} \item{n}{Number of initial observations.} } \note{This function is not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/Jfunctions.Rd0000644000176200001440000000470314130622661015576 0ustar liggesusers\name{Jfunctions} \alias{Jfunctions} \alias{J00} \alias{J10} \alias{J11} \alias{J20} \title{Numerical Routine J and Some Derivatives} \description{ J00 represents the function \eqn{J(x, y, v),} where for real numbers \eqn{x, y} and \eqn{v \in [0, 1],} \deqn{J(x, y, v) = \int_0^v \exp((1-t)x + ty) d t = \frac{\exp(x + v(y - x)) - \exp(x)}{y - x}.}{J(x, y, v) = int_0^v exp((1 - t) x + t y) d t = (exp(x + v(y - x)) - exp(x))/(y - x).} The functions Jab give the respective derivatives \eqn{J_{ab}} for \eqn{v = 1}, i.e. \deqn{J_{ab}(x, y) = \frac{\partial^{a+b}}{\partial x^a \partial y^b} J(x, y).}{J_{ab}(x, y) = (partial ^ {a + b}) / (\partial x ^ a \partial y ^ b) J(x, y).} Specifically, \deqn{J_{10}(x, y) = \frac{\exp(y) - \exp(x) - (y - x) \exp(x)}{(y - x)^2};}{J_{10}(x, y) = (exp(y) - exp(x) - (y - x) exp(x))/((y - x) ^ 2);} \deqn{J_{11}(x, y) = \frac{(y - x)(\exp(x) + \exp(y)) + 2 (\exp(y) - \exp(x))}{(y - x)^3};}{J_{11}(x, y) = ((y - x)(exp(x) + exp(y)) + 2 (exp(y) - exp(x)))/((y - x) ^ 3);} \deqn{J_{20}(x, y) = 2\frac{\exp(y) - \exp(x) - (y - x)\exp(x)-(y - x)^2 \exp(x)}{(y - x)^3}.}{J_{20}(x, y) = 2(exp(y) - exp(x) - (y - x) exp(x) - (y - x) ^ 2 exp(x)) / ((y - x) ^ 3).} } \usage{ J00(x, y, v) J10(x, y) J11(x, y) J20(x, y)} \arguments{ \item{x}{Vector of length \eqn{d} with real entries.} \item{y}{Vector of length \eqn{d} with real entries.} \item{v}{Number in \eqn{[0, 1]^d}.} } \value{Value of the respective function.} \note{Taylor approximations are used if \eqn{y-x} is small. We refer to Duembgen et al (2011, Section 6) for details. These functions are not intended to be invoked by the end user.} \references{ Duembgen, L, Huesler, A. and Rufibach, K. (2010) Active set and EM algorithms for log-concave densities based on complete and censored data. Technical report 61, IMSV, Univ. of Bern, available at \url{https://arxiv.org/abs/0707.4643}. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/rlogcon.Rd0000644000176200001440000000432614130622661015120 0ustar liggesusers\name{rlogcon} \Rdversion{1.1} \alias{rlogcon} \title{Generate random sample from the log-concave and the smoothed log-concave density estimator} \description{Generate a random sample from a distribution with density \eqn{\hat f_n} and \eqn{\hat f_n^*}, as described in Duembgen and Rufibach (2009, Section 3).} \usage{rlogcon(n, x0)} \arguments{ \item{n}{Size of random sample to be generated.} \item{x0}{Sorted vector of independent and identically distributed numbers, not necessarily unique.} } \value{ \item{X}{Random sample from \eqn{\hat f_n}.} \item{X_star}{Random sample from \eqn{\hat f_n^*}.} \item{U}{Uniform random sample of size \code{n} used in the generation of \code{X}.} \item{Z}{Normal random sample of size \code{n} used in the generation of \code{X_star}.} \item{f}{Computed log-concave density estimator.} \item{f.smoothed}{List containing smoothed log-concave density estimator, as output by \code{\link{evaluateLogConDens}}.} \item{x}{Vector of distinct observations generated from \code{x0}.} \item{w}{Weights corresponding to \code{x}.} } \references{ Duembgen, L. and Rufibach, K. (2009) Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. \emph{Bernoulli}, \bold{15(1)}, 40--68. Duembgen, L. and Rufibach, K. (2011) logcondens: Computations Related to Univariate Log-Concave Density Estimation. \emph{Journal of Statistical Software}, \bold{39(6)}, 1--28. \doi{https://doi.org/10.18637/jss.v039.i06} } \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html} } \examples{ ## =================================================== ## Generate random samples as described in Section 3 of ## Duembgen and Rufibach (2009) ## =================================================== x0 <- rnorm(111) n <- 22 random <- rlogcon(n, x0) ## sample of size n from the log-concave density estimator random$X ## sample of size n from the smoothed log-concave density estimator random$X_star } \keyword{htest} \keyword{nonparametric} logcondens/man/reparametrizations.Rd0000644000176200001440000000262214130622661017375 0ustar liggesusers\name{reparametrizations} \alias{reparametrizations} \alias{phieta} \alias{etaphi} \title{Changes Between Parametrizations} \description{ Given a vector \eqn{(\phi_1, \ldots, \phi_m)} representing the values of a piecewise linear concave function at \eqn{x_1, \ldots, x_m,} \code{\link{etaphi}} returns a column vector with the entries \deqn{{\bold{\eta}} = \Bigl(\phi_1, \Bigl(\eta_1 + \sum_{j=2}^m (x_i-x_{i-1})\eta_i\Bigr)_{i=2}^m\Bigr).}{\eta = (\phi_1, (\eta_1 + \sum_{j=2}^m (x_i-x_{i-1})\eta_i)_{i=2}^m).} The function \code{\link{phieta}} returns a vector with the entries \deqn{{\bold{\phi}} = \Bigl(\eta_1, \Bigl(\frac{\phi_i-\phi_{i-1}}{x_i-x_{i-1}}\Bigr)_{i=2}^m\Bigr).}{\phi = (\eta_1, ((\phi_i-\phi_{i-1})/(x_i-x_{i-1}))_{i=2}^m).} } \usage{etaphi(x, eta) phieta(x, phi)} \arguments{ \item{x}{Vector of independent and identically distributed numbers, with strictly increasing entries.} \item{eta}{Vector with entries \eqn{\eta_i = \eta(x_i).}} \item{phi}{Vector with entries \eqn{\phi_i = \phi(x_i).}} } \note{These functions are not intended to be invoked by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/man/qloglin.Rd0000644000176200001440000000260414130622661015117 0ustar liggesusers\name{qloglin} \alias{qloglin} \title{Quantile Function In a Simple Log-Linear model} \description{Suppose the random variable \eqn{X} has density function \deqn{g_\theta(x) = \frac{\theta \exp(\theta x)}{\exp(\theta) - 1}}{g_\theta(x) = (\theta exp(\theta x))/(exp(\theta) - 1)} for an arbitrary real number \eqn{\theta} and \eqn{x \in [0,1]}. The function \code{\link{qloglin}} is simply the quantile function \deqn{G^{-1}_\theta(u) = \theta^{-1} \log \Big( 1 + (e^\theta - 1)u \Big)}{G^{-1}_\theta(u) = \theta^{-1} log (1 + (e^\theta - 1)u)} in this model, for \eqn{u \in [0,1]}. This quantile function is used for the computation of quantiles of \eqn{\widehat F_m} in \code{\link{quantilesLogConDens}}.} \usage{qloglin(u, t)} \arguments{ \item{u}{Vector in \eqn{[0,1]^d} where quantiles are to be computed at.} \item{t}{Parameter \eqn{\theta}.} } \value{\item{z}{Vector containing the quantiles \eqn{G_n^{-1}(u_i)} for \eqn{i = 1, \ldots, d}.}} \note{Taylor approximation is used if \eqn{\theta} is small. This function is not intended to be called by the end user.} \author{ Kaspar Rufibach, \email{kaspar.rufibach@gmail.com}, \cr \url{http://www.kasparrufibach.ch} Lutz Duembgen, \email{duembgen@stat.unibe.ch}, \cr \url{https://www.imsv.unibe.ch/about_us/staff/prof_dr_duembgen_lutz/index_eng.html}} \keyword{htest} \keyword{nonparametric} logcondens/DESCRIPTION0000644000176200001440000000323714131330403014110 0ustar liggesusersPackage: logcondens Type: Package Title: Estimate a Log-Concave Probability Density from Iid Observations Version: 2.1.6 Date: 2021-10-10 Author: Kaspar Rufibach and Lutz Duembgen Maintainer: Kaspar Rufibach Depends: R (>= 2.10) Imports: ks, graphics, stats Description: Given independent and identically distributed observations X(1), ..., X(n), compute the maximum likelihood estimator (MLE) of a density as well as a smoothed version of it under the assumption that the density is log-concave, see Rufibach (2007) and Duembgen and Rufibach (2009). The main function of the package is 'logConDens' that allows computation of the log-concave MLE and its smoothed version. In addition, we provide functions to compute (1) the value of the density and distribution function estimates (MLE and smoothed) at a given point (2) the characterizing functions of the estimator, (3) to sample from the estimated distribution, (5) to compute a two-sample permutation test based on log-concave densities, (6) the ROC curve based on log-concave estimates within cases and controls, including confidence intervals for given values of false positive fractions (7) computation of a confidence interval for the value of the true density at a fixed point. Finally, three datasets that have been used to illustrate log-concave density estimation are made available. 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QQd~$CgVCC7e`Du_:. ̂B^>Ĵ'KYPoөzȂ6PU4Mņ }ٰ,($q`D|7 '@O40c5dAIvk2Jf8 ^Uom͂B4Ȃs&n߬,(I9Eb;uYPZ,(C3J}CIAUqw0^ú,(rixئKn~˦xZc5T0$<=$x#R&Ȧ|{m<송 5얨1n`\d8P6>Go@CC: b>}X AcA<)/9 b.Wѫh data(cars) % R> fm <- lm(dist ~ speed, data = log(cars)) % R> summary(fm) % \end{CodeInput} % \begin{CodeOutput} % Call: % lm(formula = dist ~ speed, data = log(cars)) % % Residuals: % Min 1Q Median 3Q Max % -1.00215 -0.24578 -0.02898 0.20717 0.88289 % % Coefficients: % Estimate Std. Error t value Pr(>|t|) % (Intercept) -0.7297 0.3758 -1.941 0.0581 . % speed 1.6024 0.1395 11.484 2.26e-15 *** % --- % Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 % % Residual standard error: 0.4053 on 48 degrees of freedom % Multiple R-Squared: 0.7331, Adjusted R-squared: 0.7276 % F-statistic: 131.9 on 1 and 48 DF, p-value: 2.259e-15 % \end{CodeOutput} % \end{CodeChunk} % \end{verbatim} % If you prepare your paper using \foopkg{Sweave} (which is recommended % if you describe an \textsf{R} package) do \emph{not} include % |Sweave.sty| into your document, the necessary commands are already available within % |jss.cls|. To prevent \foopkg{Sweave} from including |Sweave.sty| % automatically you need to include a line like % \begin{verbatim} % %% need no \usepackage{Sweave.sty} % \end{verbatim} % (with `\%\%') into the header of your document. % % If this basic infrastructure for typesetting your code is not % sufficient, you can also use other \LaTeX{} packages like the % \foopkg{listings} package. % % \subsubsection*{Mathematical formulae} % % Commonly used operators like $\mathsf{E}$, $\mathsf{VAR}$, $\mathsf{COV}$, and $\mathsf{P}$ should be set % using the commands |\E|, |\VAR|, |\COV| and |\Prob|. Beyond this, \foopkg{jss} does not % provide (or enforce) a certain mathematical notation. However, using the \foopkg{statex} % package (e.g., available from CTAN and in \TeX{} Live) could be useful. % % % \section{Using pdf\TeX{} and \textbf{\sc Bib}\TeX{}} \label{sec:TeX} % % \subsubsection*{Using pdf\TeX{}} % % A \LaTeX{} document (|foo.tex|, say) using |jss.cls| needs to be compiled using % pdf\TeX{}, typically this will be done using either of the % following commands: % \begin{verbatim} % pdflatex foo.tex % % texi2dvi --pdf foo.tex % % texi2pdf foo.tex % \end{verbatim} % If you are not using command line tools but some integrated GUI editor for % \LaTeX{} documents you will have to press the `pdf\LaTeX{}' button % (as opposed to the `\LaTeX{}' button). % % All graphics included into the document have to be in a format pdf\TeX{} can % deal with, i.e., PDF for vector graphics or JPG/PNG/etc. for bitmaps/raster graphics. % If you cannot produce PDF graphics directly but only PS/EPS, these can % be converted using |ps2pdf| or |epstopdf| (usually preferred). % % \emph{Hint.} If you are used to compiling your documents with standard \LaTeX{} % and then getting automatic reloads of the resulting DVI document % in your DVI viewer, which is not possible with PDF documents in many % PDF viewers: you might want to look at \foopkg{xpdf} (Linux) or \foopkg{gsview} % (Windows, see \url{http://www.cs.wisc.edu/~ghost/gsview/}) % which have a reload function. % % \emph{Hint.} If you want to use markup in section headers you will usually % have to escape it for the PDF bookmarks by giving the text for the % bookmark explicitly without markup, e.g., % \begin{verbatim} % \section[Calling C++ from R]{Calling \proglang{C++} from \proglang{R}} % \end{verbatim} % % \emph{Hint.} If you know how to produce \LaTeX{} documents that can be % processed with both \LaTeX{} and pdf\TeX{}, you can do so if you provide % an EPS substitute for |jsslogo.jpg| (e.g. an empty or converted |jsslogo.eps|). % Note, however, that the final document needs to be processed with pdf\TeX{}. % Neither this manual nor the JSS encourage or support compilation of % JSS documents with standard \LaTeX{}. % % % \subsubsection*{References with \textbf{\sc Bib}\TeX{}} % % The format for references (e.g., articles, books, software, proceedings) % should look like this % % \begin{quote} % Brown RL, Durbin J, Evans JM (1975). % \newblock \enquote{Techniques for Testing the Constancy of Regression % Relationships over Time.} % \newblock \emph{Journal of the Royal Statistical Society B}, \textbf{37}, % 149--163. % % Friendly M (2000). % \newblock \emph{Visualizing Categorical Data}. % \newblock SAS Insitute, Carey, NC. % % {\textsf{R} Development Core Team} (2004). % \newblock \emph{\textsf{R}: {A} Language and Environment for Statistical % Computing}. % \newblock \textsf{R} Foundation for Statistical Computing, Vienna, Austria. % \newblock {ISBN} 3-900051-00-3, URL~\url{http://www.R-project.org/}. % % Urbanek S, Theus M (2003). % \newblock \enquote{\foopkg{iPlots} -- {H}igh Interaction Graphics for \textsf{R}.} % \newblock In K~Hornik, F~Leisch, A~Zeileis (eds.), \enquote{Proceedings of the % 3rd International Workshop on Distributed Statistical Computing, Vienna, % Austria,} {ISSN 1609-395X}, % URL~\url{http://www.ci.tuwien.ac.at/Conferences/DSC-2003/Proceedings/}. % \end{quote} % % \emph{Important.} Note, that also the titles of papers are in title style % (as opposed to sentence style), i.e., they are capitalized. % The first word after a colon `:' is always capitalized. Furthermore, commands % like \verb/\proglang/ and \verb/\pkg/ should also be used for the % references. The names of journals or proceeding volumes should not % be abbreviated. % % The easiest way to achieve this % is to use \textsc{Bib}\TeX{} together with the style file |jss.bst|. % To do so, the references just have to be included in a \textsc{Bib}\TeX{} file, % |foo.bib| say, which has to be included at the end of the \LaTeX{} % document by |\bibliography{foo}|. % Note, that to obtain references in the format above, the |title| field % in your bib file, needs to be capitalized (contrary to the folklore, % there are \textsc{Bib}\TeX{} styles that rely on this even for |@Article| % entries), i.e. the entry |title = {Visualizing Categorical Data}| is % correct, while entries like |title = {Visualizing categorical data}| % or (even worse) |title = {{Visualizing categorical data}}| are not. % % The default in |jss.cls| is to use the \foopkg{natbib} package % with options |authoryear|, |round| and |longnamesfirst|. If you cite % any article with six or more authors the latter option should be % turned off. This can be done by using the option |shortnames| % when loading the |jss.cls| class % \begin{verbatim} % \documentclass[article,shortnames]{jss} % \end{verbatim} % % % %\newpage % % \section{The code} \label{sec:code} % % \subsection{The batch file} % % First comes the code for creating the batch file \file{\filename.ins} % which in turn can be used for producing the package and driver files. % % \begin{macrocode} %<*install> \begin{filecontents}{\filename.ins} % Simply TeX or LaTeX this file to extract various files from the source % file `jss.dtx'. \def\filedate{2004/09/29} \def\batchfile{jss.ins} \input docstrip.tex \generateFile{jss.drv}{t}{\from{jss.dtx}{driver}} \generateFile{jss.cls}{t}{\from{jss.dtx}{class}} \Msg{*******************************************************} \Msg{* For documentation, run LaTeX on jss.dtx or jss.drv. *} \Msg{*******************************************************} \end{filecontents} % % \end{macrocode} % % \subsection{The driver} % % Next comes the documentation driver file for \TeX{}, i.e., the file % that will produce the documentation you are currently reading. It % will be extracted from this file by the \texttt{docstrip} % program. Since it is the first code in the file one can % alternatively process this file directly with \LaTeXe{} to obtain % the documentation. % % \begin{macrocode} %<*driver> \documentclass[a4paper]{ltxdoc} \providecommand{\file}[1]{\texttt{#1}} \providecommand{\pkg}[1]{{\fontseries{b}\selectfont #1}} \usepackage{color,hyperref,a4wide} \oddsidemargin1.2cm \textwidth14.2cm \textheight23.3cm \topmargin-.7cm \setlength{\parskip}{0.7ex plus0.1ex minus0.1ex} \setlength{\parindent}{0em} \begin{document} \OnlyDescription \DocInput{jss.dtx} \end{document} % % \end{macrocode} % % \subsection{The class} % % Next is the main part, the code for the class file. % % It requires \LaTeXe{} % \begin{macrocode} %<*class> \NeedsTeXFormat{LaTeX2e} \ProvidesClass{jss}[\filedate\space\fileversion\space jss class by Achim Zeileis] % % \end{macrocode} % and is based on the \texttt{article} class. But before we load % the class we declare and process some options. % These reflects wether we want to write an article, code snippet, % a book review or software review. The \texttt{shortnames} option % is for loading \texttt{natbib} \emph{without} the option % \texttt{longnamesfirst}. The \texttt{nojss} option suppresses JSS header and footer. % \begin{macrocode} %<*class> %% options \newif\if@article \newif\if@codesnippet \newif\if@bookreview \newif\if@softwarereview \newif\if@review \newif\if@shortnames \newif\if@nojss \@articletrue \@codesnippetfalse \@bookreviewfalse \@softwarereviewfalse \@reviewfalse \@shortnamesfalse \@nojssfalse \DeclareOption{article}{\@articletrue% \@codesnippetfalse \@bookreviewfalse \@softwarereviewfalse} \DeclareOption{codesnippet}{\@articlefalse% \@codesnippettrue \@bookreviewfalse \@softwarereviewfalse} \DeclareOption{bookreview}{\@articlefalse% \@codesnippetfalse \@bookreviewtrue \@softwarereviewfalse} \DeclareOption{softwarereview}{\@articlefalse% \@codesnippetfalse \@bookreviewfalse \@softwarereviewtrue} \DeclareOption{shortnames}{\@shortnamestrue} \DeclareOption{nojss}{\@nojsstrue} \ProcessOptions \LoadClass[11pt,a4paper,twoside]{article} % % \end{macrocode} % % A few packages are required and the font encoding is specified. % \begin{macrocode} %<*class> %% required packages \RequirePackage{graphicx,a4wide,color,ae,fancyvrb} \RequirePackage[T1]{fontenc} \IfFileExists{upquote.sty}{\RequirePackage{upquote}}{} % % \end{macrocode} % % In addition, \texttt{hyperref} is included later on. % The bibliography is generated using \texttt{natbib} and % the \textsc{Bib}\TeX{} style \file{jss.bst}. % \begin{macrocode} %<*class> %% bibliography \if@shortnames \usepackage[authoryear,round]{natbib} \else \usepackage[authoryear,round,longnamesfirst]{natbib} \fi \bibpunct{(}{)}{;}{a}{}{,} \bibliographystyle{jss} % % \end{macrocode} % % Paragraphs are not indented, instead \verb/\parskip/ is % increased. % \begin{macrocode} %<*class> %% paragraphs \setlength{\parskip}{0.7ex plus0.1ex minus0.1ex} \setlength{\parindent}{0em} % % \end{macrocode} % % To process the meta information we need some new commands: % for all publications, % \begin{macrocode} %<*class> %% for all publications \newcommand{\Address}[1]{\def\@Address{#1}} \newcommand{\Plaintitle}[1]{\def\@Plaintitle{#1}} \newcommand{\Shorttitle}[1]{\def\@Shorttitle{#1}} \newcommand{\Plainauthor}[1]{\def\@Plainauthor{#1}} \newcommand{\Volume}[1]{\def\@Volume{#1}} \newcommand{\Year}[1]{\def\@Year{#1}} \newcommand{\Month}[1]{\def\@Month{#1}} \newcommand{\Issue}[1]{\def\@Issue{#1}} \newcommand{\Submitdate}[1]{\def\@Submitdate{#1}} % % \end{macrocode} % for articles and code snippets, % \begin{macrocode} %<*class> %% for articles and code snippets \newcommand{\Acceptdate}[1]{\def\@Acceptdate{#1}} \newcommand{\Abstract}[1]{\def\@Abstract{#1}} \newcommand{\Keywords}[1]{\def\@Keywords{#1}} \newcommand{\Plainkeywords}[1]{\def\@Plainkeywords{#1}} % % \end{macrocode} % for book and software reviews, % \begin{macrocode} %<*class> %% for book and software reviews \newcommand{\Reviewer}[1]{\def\@Reviewer{#1}} \newcommand{\Booktitle}[1]{\def\@Booktitle{#1}} \newcommand{\Bookauthor}[1]{\def\@Bookauthor{#1}} \newcommand{\Publisher}[1]{\def\@Publisher{#1}} \newcommand{\Pubaddress}[1]{\def\@Pubaddress{#1}} \newcommand{\Pubyear}[1]{\def\@Pubyear{#1}} \newcommand{\ISBN}[1]{\def\@ISBN{#1}} \newcommand{\Pages}[1]{\def\@Pages{#1}} \newcommand{\Price}[1]{\def\@Price{#1}} \newcommand{\Plainreviewer}[1]{\def\@Plainreviewer{#1}} \newcommand{\Softwaretitle}[1]{\def\@Softwaretitle{#1}} \newcommand{\URL}[1]{\def\@URL{#1}} % % \end{macrocode} % and for internal use only. % \begin{macrocode} %<*class> %% for internal use \newcommand{\Seriesname}[1]{\def\@Seriesname{#1}} \newcommand{\Hypersubject}[1]{\def\@Hypersubject{#1}} \newcommand{\Hyperauthor}[1]{\def\@Hyperauthor{#1}} \newcommand{\Footername}[1]{\def\@Footername{#1}} \newcommand{\Firstdate}[1]{\def\@Firstdate{#1}} \newcommand{\Seconddate}[1]{\def\@Seconddate{#1}} \newcommand{\Reviewauthor}[1]{\def\@Reviewauthor{#1}} % % \end{macrocode} % % Some defaults for theses commands are specified, which % are (hopefully) a useful guidance when using the % \file{\filename.cls}. % \begin{macrocode} %<*class> %% defaults \author{Firstname Lastname\\Affiliation} \title{Title} \Abstract{---!!!---an abstract is required---!!!---} \Plainauthor{\@author} \Volume{VV} \Year{YYYY} \Month{MMMMMM} \Issue{II} \Submitdate{yyyy-mm-dd} \Acceptdate{yyyy-mm-dd} \Address{ Firstname Lastname\\ Affiliation\\ Address, Country\\ E-mail: \email{name@address}\\ URL: \url{http://link/to/webpage/} } \Reviewer{Firstname Lastname\\Affiliation} \Plainreviewer{Firstname Lastname} \Booktitle{Book Title} \Bookauthor{Book Author} \Publisher{Publisher} \Pubaddress{Publisher's Address} \Pubyear{YYY} \ISBN{x-xxxxx-xxx-x} \Pages{xv + 123} \Price{USD 69.95 (P)} \URL{http://link/to/webpage/} % % \end{macrocode} % % Conditional on the type of document several other defaults % and some meta information is stored. % \begin{macrocode} %<*class> \if@article \Seriesname{Issue} \Hypersubject{Journal of Statistical Software} \Plaintitle{\@title} \Shorttitle{\@title} \Plainkeywords{\@Keywords} \fi \if@codesnippet \Seriesname{Code Snippet} \Hypersubject{Journal of Statistical Software -- Code Snippets} \Plaintitle{\@title} \Shorttitle{\@title} \Plainkeywords{\@Keywords} \fi \if@bookreview \Seriesname{Book Review} \Hypersubject{Journal of Statistical Software -- Book Reviews} \Plaintitle{\@Booktitle} \Shorttitle{\@Booktitle} \Reviewauthor{\@Bookauthor\\ \@Publisher, \@Pubaddress, \@Pubyear.\\ ISBN~\@ISBN. \@Pages~pp. \@Price.\\ \url{\@URL}} \Plainkeywords{} \@reviewtrue \fi \if@softwarereview \Seriesname{Software Review} \Hypersubject{Journal of Statistical Software -- Software Reviews} \Plaintitle{\@Softwaretitle} \Shorttitle{\@Softwaretitle} \Booktitle{\@Softwaretitle} \Reviewauthor{\@Publisher, \@Pubaddress. \@Price.\\ \url{\@URL}} \Plainkeywords{} \@reviewtrue \fi \if@review \Hyperauthor{\@Plainreviewer} \Keywords{} \Footername{Reviewer} \Firstdate{\textit{Published:} \@Submitdate} \Seconddate{} \else \Hyperauthor{\@Plainauthor} \Keywords{---!!!---at least one keyword is required---!!!---} \Footername{Affiliation} \Firstdate{\textit{Submitted:} \@Submitdate} \Seconddate{\textit{Accepted:} \@Acceptdate} \fi % % \end{macrocode} % % For typesetting of code some basic infrastructure along % the lines of Sweave is provided. First, the Sweave commands % are provided explicitly, % \begin{macrocode} %<*class> %% Sweave(-like) \DefineVerbatimEnvironment{Sinput}{Verbatim}{fontshape=sl} \DefineVerbatimEnvironment{Soutput}{Verbatim}{} \DefineVerbatimEnvironment{Scode}{Verbatim}{fontshape=sl} \newenvironment{Schunk}{}{} % % \end{macrocode} % and analogous commands with more neutral names for general % pieces of code. % \begin{macrocode} %<*class> \DefineVerbatimEnvironment{Code}{Verbatim}{} \DefineVerbatimEnvironment{CodeInput}{Verbatim}{fontshape=sl} \DefineVerbatimEnvironment{CodeOutput}{Verbatim}{} \newenvironment{CodeChunk}{}{} \setkeys{Gin}{width=0.8\textwidth} % % \end{macrocode} % % The header and footer of JSS publications displays the logo, % the publication information and some further links. Here, % we define the footer first (because it must be included % before \texttt{hyperref} in {\TeX}live). It contains the somewhat extended % publication information (from the header), preceeded by the address of the % author/reviewer. % \begin{macrocode} %<*class> %% footer \newlength{\footerskip} \setlength{\footerskip}{2.5\baselineskip plus 2ex minus 0.5ex} \newcommand{\makefooter}{% \vspace{\footerskip} \if@nojss \begin{samepage} \textbf{\large \@Footername: \nopagebreak}\\[.3\baselineskip] \nopagebreak \@Address \nopagebreak \end{samepage} \else \begin{samepage} \textbf{\large \@Footername: \nopagebreak}\\[.3\baselineskip] \nopagebreak \@Address \nopagebreak \vfill \hrule \nopagebreak \vspace{.1\baselineskip} {\fontfamily{pzc} \fontsize{13}{15} \selectfont Journal of Statistical Software} \hfill \url{http://www.jstatsoft.org/}\\ \nopagebreak published by the American Statistical Association \hfill \url{http://www.amstat.org/}\\[.3\baselineskip] \nopagebreak {Volume~\@Volume, \@Seriesname~\@Issue} \hfill \@Firstdate\\ \nopagebreak {\@Month{} \@Year} \hfill \@Seconddate \nopagebreak \vspace{.3\baselineskip} \hrule \end{samepage} \fi } % % \end{macrocode} % % We include the footer at the end of the document. % \begin{macrocode} %<*class> \AtEndDocument{\makefooter} % % \end{macrocode} % % After defining this, we can require the \texttt{hyperref} package. % \begin{macrocode} %<*class> %% required packages \RequirePackage{hyperref} % % \end{macrocode} % and proceed to define the header. % % The header for all JSS publications has the logo \file{jsslogo.jpg} % along with the publication information. % \begin{macrocode} %<*class> %% new \maketitle \def\@myoddhead{ {\color{white} JSS}\\[-1.42cm] \hspace{-2em} \includegraphics[height=23mm,keepaspectratio]{jsslogo} \hfill \parbox[b][23mm]{118mm}{\hrule height 3pt \center{ {\fontfamily{pzc} \fontsize{28}{32} \selectfont Journal of Statistical Software} \vfill {\it \small \@Month{} \@Year, Volume~\@Volume, \@Seriesname~\@Issue.% \hfill \href{http://www.jstatsoft.org/}{http://www.jstatsoft.org/}}}\\[0.1cm] \hrule height 3pt}} % % \end{macrocode} % % This header is then used in the re-defined \verb/\maketitle/: % \begin{macrocode} %<*class> \if@review \renewcommand{\maketitle}{ \if@nojss %% \@oddhead{\@myoddhead}\\[3\baselineskip] \else \@oddhead{\@myoddhead}\\[3\baselineskip] \fi {\large \noindent Reviewer: \@Reviewer \vspace{\baselineskip} \hrule \vspace{\baselineskip} \textbf{\@Booktitle} \begin{quotation} \noindent \@Reviewauthor \end{quotation} \vspace{0.7\baselineskip} \hrule \vspace{1.3\baselineskip} } \thispagestyle{empty} \if@nojss \markboth{\centerline{\@Shorttitle}}{\centerline{\@Hyperauthor}} \else \markboth{\centerline{\@Shorttitle}}{\centerline{\@Hypersubject}} \fi \pagestyle{myheadings} } \else \def\maketitle{ \if@nojss %% \@oddhead{\@myoddhead} \par \else \@oddhead{\@myoddhead} \par \fi \begingroup \def\thefootnote{\fnsymbol{footnote}} \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}} \long\def\@makefntext##1{\parindent 1em\noindent \hbox to1.8em{\hss $\m@th ^{\@thefnmark}$}##1} \@maketitle \@thanks \endgroup \setcounter{footnote}{0} \thispagestyle{empty} \if@nojss \markboth{\centerline{\@Shorttitle}}{\centerline{\@Hyperauthor}} \else \markboth{\centerline{\@Shorttitle}}{\centerline{\@Hypersubject}} \fi \pagestyle{myheadings} \let\maketitle\relax \let\@maketitle\relax \gdef\@thanks{}\gdef\@author{}\gdef\@title{}\let\thanks\relax } \def\@maketitle{\vbox{\hsize\textwidth \linewidth\hsize \if@nojss %% \vskip 1in \else \vskip 1in \fi {\centering {\LARGE\bf \@title\par} \vskip 0.2in plus 1fil minus 0.1in { \def\and{\unskip\enspace{\rm and}\enspace}% \def\And{\end{tabular}\hss \egroup \hskip 1in plus 2fil \hbox to 0pt\bgroup\hss \begin{tabular}[t]{c}\large\bf\rule{\z@}{24pt}\ignorespaces}% \def\AND{\end{tabular}\hss\egroup \hfil\hfil\egroup \vskip 0.1in plus 1fil minus 0.05in \hbox to \linewidth\bgroup\rule{\z@}{10pt} \hfil\hfil \hbox to 0pt\bgroup\hss \begin{tabular}[t]{c}\large\bf\rule{\z@}{24pt}\ignorespaces} \hbox to \linewidth\bgroup\rule{\z@}{10pt} \hfil\hfil \hbox to 0pt\bgroup\hss \begin{tabular}[t]{c}\large\bf\rule{\z@}{24pt}\@author \end{tabular}\hss\egroup \hfil\hfil\egroup} \vskip 0.3in minus 0.1in \hrule \begin{abstract} \@Abstract \end{abstract}} \textit{Keywords}:~\@Keywords. \vskip 0.1in minus 0.05in \hrule \vskip 0.2in minus 0.1in }} \fi % % \end{macrocode} % % The appearance of sections, subsections and subsubsections is % controlled by % \begin{macrocode} %<*class> %% sections, subsections, and subsubsections \newlength{\preXLskip} \newlength{\preLskip} \newlength{\preMskip} \newlength{\preSskip} \newlength{\postMskip} \newlength{\postSskip} \setlength{\preXLskip}{1.8\baselineskip plus 0.5ex minus 0ex} \setlength{\preLskip}{1.5\baselineskip plus 0.3ex minus 0ex} \setlength{\preMskip}{1\baselineskip plus 0.2ex minus 0ex} \setlength{\preSskip}{.8\baselineskip plus 0.2ex minus 0ex} \setlength{\postMskip}{.5\baselineskip plus 0ex minus 0.1ex} \setlength{\postSskip}{.3\baselineskip plus 0ex minus 0.1ex} \newcommand{\jsssec}[2][default]{\vskip \preXLskip% \pdfbookmark[1]{#1}{Section.\thesection.#1}% \refstepcounter{section}% \centerline{\textbf{\Large \thesection. #2}} \nopagebreak \vskip \postMskip \nopagebreak} \newcommand{\jsssecnn}[1]{\vskip \preXLskip% \centerline{\textbf{\Large #1}} \nopagebreak \vskip \postMskip \nopagebreak} \newcommand{\jsssubsec}[2][default]{\vskip \preMskip% \pdfbookmark[2]{#1}{Subsection.\thesubsection.#1}% \refstepcounter{subsection}% \textbf{\large \thesubsection. #2} \nopagebreak \vskip \postSskip \nopagebreak} \newcommand{\jsssubsecnn}[1]{\vskip \preMskip% \textbf{\large #1} \nopagebreak \vskip \postSskip \nopagebreak} \newcommand{\jsssubsubsec}[2][default]{\vskip \preSskip% \pdfbookmark[3]{#1}{Subsubsection.\thesubsubsection.#1}% \refstepcounter{subsubsection}% {\large \textit{#2}} \nopagebreak \vskip \postSskip \nopagebreak} \newcommand{\jsssubsubsecnn}[1]{\vskip \preSskip% {\textit{\large #1}} \nopagebreak \vskip \postSskip \nopagebreak} \newcommand{\jsssimplesec}[2][default]{\vskip \preLskip% %% \pdfbookmark[1]{#1}{Section.\thesection.#1}% \refstepcounter{section}% \textbf{\large #1} \nopagebreak \vskip \postSskip \nopagebreak} \newcommand{\jsssimplesecnn}[1]{\vskip \preLskip% \textbf{\large #1} \nopagebreak \vskip \postSskip \nopagebreak} \if@review \renewcommand{\section}{\secdef \jsssimplesec \jsssimplesecnn} \renewcommand{\subsection}{\secdef \jsssimplesec \jsssimplesecnn} \renewcommand{\subsubsection}{\secdef \jsssimplesec \jsssimplesecnn} \else \renewcommand{\section}{\secdef \jsssec \jsssecnn} \renewcommand{\subsection}{\secdef \jsssubsec \jsssubsecnn} \renewcommand{\subsubsection}{\secdef \jsssubsubsec \jsssubsubsecnn} \fi % % \end{macrocode} % % The hypersetup uses some modified colors % \begin{macrocode} %<*class> %% colors \definecolor{Red}{rgb}{0.5,0,0} \definecolor{Blue}{rgb}{0,0,0.5} % % \end{macrocode} % and is then defined by % \begin{macrocode} %<*class> \if@review \hypersetup{% hyperindex = {true}, colorlinks = {true}, linktocpage = {true}, plainpages = {false}, linkcolor = {Blue}, citecolor = {Blue}, urlcolor = {Red}, pdfstartview = {Fit}, pdfpagemode = {None}, pdfview = {XYZ null null null} } \else \hypersetup{% hyperindex = {true}, colorlinks = {true}, linktocpage = {true}, plainpages = {false}, linkcolor = {Blue}, citecolor = {Blue}, urlcolor = {Red}, pdfstartview = {Fit}, pdfpagemode = {UseOutlines}, pdfview = {XYZ null null null} } \fi % % \end{macrocode} % % The information for the hyper summary requires % some information which has not been processed % before the beginning of the document. Therefore, % we need a second \verb/\hypersetup/. % \begin{macrocode} %<*class> \if@nojss \AtBeginDocument{ \hypersetup{% pdfauthor = {\@Hyperauthor}, pdftitle = {\@Plaintitle}, pdfkeywords = {\@Plainkeywords} } } \else \AtBeginDocument{ \hypersetup{% pdfauthor = {\@Hyperauthor}, pdftitle = {\@Plaintitle}, pdfsubject = {\@Hypersubject}, pdfkeywords = {\@Plainkeywords} } } \fi % % \end{macrocode} % % We put the header at the beginning of the % document (for footer see above). % \begin{macrocode} %<*class> \AtBeginDocument{\maketitle} % % \end{macrocode} % % Finally, some additional commands are provided for writing about % software (code, programming languages, packages), % \begin{macrocode} %<*class> %% commands \makeatletter \newcommand\code{\bgroup\@makeother\_\@makeother\~\@makeother\$\@codex} \def\@codex#1{{\normalfont\ttfamily\hyphenchar\font=-1 #1}\egroup} \makeatother %%\let\code=\texttt \let\proglang=\textsf \newcommand{\pkg}[1]{{\fontseries{b}\selectfont #1}} % % \end{macrocode} % for specifying e-mail addresses, % \begin{macrocode} %<*class> \newcommand{\email}[1]{\href{mailto:#1}{\normalfont\texttt{#1}}} % % \end{macrocode} % digital object identifiers (DOIs), % \begin{macrocode} %<*class> \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\normalfont\texttt{doi:#1}}} % % \end{macrocode} % and for mathematical notation. % \begin{macrocode} %<*class> \newcommand{\E}{\mathsf{E}} \newcommand{\VAR}{\mathsf{VAR}} \newcommand{\COV}{\mathsf{COV}} \newcommand{\Prob}{\mathsf{P}} % % \end{macrocode} logcondens/vignettes/results/0000755000176200001440000000000014131325170016113 5ustar liggesuserslogcondens/vignettes/results/pvals_setting=3_n1=20_n2=25_tau=1.5.txt0000644000176200001440000010235512615675116024633 0ustar liggesusers"logcon","smooth","K.S" "1",0.002,NA,0.000995587375361429 "2",0.651,NA,0.990625606021044 "3",0.535,NA,0.89860822198259 "4",0.14,NA,0.195517517300654 "5",0.159,NA,0.104610577876392 "6",0.886,NA,0.959703183178312 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Wellner}, title = {Nonparametric Estimation of Multivariate Convex-Transformed Densities}, journal = {The Annals of Statistics}, year = {2010}, note = {To appear} } @ARTICLE{silverman_82, author = {Silverman, B. W.}, title = {On the Estimation of a Probability Density Function by the Maximum Penalized Likelihood Method}, journal = {The Annals of Statistics}, year = {1982}, volume = {10}, pages = {795--810}, number = {3}, fjournal = {The Annals of Statistics}, issn = {0090-5364} } @MANUAL{R, title = {\proglang{R}: A Language and Environment for Statistical Computing}, author = {\proglang{R} Development Core Team}, organization = {\proglang{R} Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2010}, note = {{ISBN} 3-900051-07-0}, owner = {rufibach}, timestamp = {2010.05.31}, url = {http://www.R-project.org} } @ARTICLE{walther_08, author = {Walther, Guenther}, title = {Inference and Modeling with Log-concave Distributions}, journal = {Statistical Science}, year = {2009}, volume = {24}, pages = {319--327}, number = {3} } @comment{jabref-meta: selector_publisher:} @comment{jabref-meta: selector_author:} @comment{jabref-meta: selector_journal:} @comment{jabref-meta: selector_keywords:} logcondens/vignettes/plots/0000755000176200001440000000000014131325167015561 5ustar liggesuserslogcondens/vignettes/plots/-plot2sample.pdf0000644000176200001440000002454514131325236020602 0ustar liggesusers%PDF-1.4 %ρ\r 1 0 obj << /CreationDate (D:20211012173246) /ModDate (D:20211012173246) /Title (R Graphics Output) /Producer (R 4.1.1) /Creator (R) >> endobj 2 0 obj << /Type /Catalog /Pages 3 0 R >> endobj 7 0 obj << /Type /Page /Parent 3 0 R /Contents 8 0 R /Resources 4 0 R >> endobj 8 0 obj << /Length 6711 /Filter /FlateDecode >> stream xݜK%IuWJ~LA6RK *`ܭFP^kGfit'#ΌΈ/o<:>jpᨡuʵ>B}֙vOdG)*V qڍbʳ<Ǧ)[sן~$O_(7.?xneԍMx{_[͟4c6}9f˯-y>r\ZN:KfmQ\)'#ys`r4/wc$rUN^~q7&GM EY>^4qq~2;_DquWq^=^{u]ȏ/S~peEhMi%ݿ.HEo_ʏ|+?|?|zWk3!S.ߒ^)tTrFCޥ,sZEum{2m۳!IulmOZl>']"ͮۀcgEJ?1'N!ЏU<Gru S$7sTe/ϺD~4Ϻ%!kG{=O$Y ?l:cDϓluR\ӗ0dS55T/#_k\AtSNIf/u||ϧ (w=*nz#Mi ep-bH*싂( SR\s-J(s-uEխ5Hf9_y2[0ע]gΦV5iբZZz)T]OV \sp=̵fz'5+9me>ŃUӤ9 fUlحR+n:۪4@ fW <5ij Uԁ0sZZ2wiA=ѢPCb0Qšf̨_AŻ6+꟮-mכ}m_iÿ4q=b~K3f'qnG} K_4eݞ,Z,|[⁷sm/q_ e92cF|GF|O2ŗgZb+:GѾQ^@km_wPGEVF@9ⓅYjd,TZ̺ #uEo/3h̡oG3 ̌u.4 = | \uJg51E2Bxn?nՏ^3>',I#odIAI:Y}HR[2N=nOV?2Uxu>vR|v'a$0Za2 YIjLK.C4GB|>JHLRz;o|IEOIa-F|: #~X:~'#_I2 lOu:P !_ѭꟘtjzח}k?¿=K7?Ѫ%!7`wMjm_G>᪶!!QοV?G~tXTjkO!'aR[SQedeX|jVZa#>ѣ _{.SBo: a>fm/!R]?1?~p|ay|< ʁ9Lwo'iC>7{(Ϩ?3Q_?q?FOi_q;GưƎ'kC5Wa| o6M5Se|\_t"eX1MM&֓&G j7d&h{#G|eh~羽Zi#L#}<ਸ਼|MywD2! 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Several algorithms have been proposed to estimate such a density. Two of those algorithms, an iterative convex minorant and an active set algorithm, are implemented in the \proglang{R} package \pkg{logcondens}. While these algorithms are discussed elsewhere, we describe in this paper the use of the \pkg{logcondens} package and discuss functions and datasets related to log-concave density estimation contained in the package. In particular, we provide functions to (1) compute the maximum likelihood estimate (MLE) as well as a smoothed log-concave density estimator derived from the MLE, (2) evaluate the estimated density, distribution and quantile functions at arbitrary points, (3) compute the characterizing functions of the MLE, (4) sample from the estimated distribution, and finally (5) perform a two-sample permutation test using a modified Kolmogorov-Smirnov test statistic. In addition, \pkg{logcondens} makes two datasets available that have been used to illustrate log-concave density estimation. } \Keywords{log-concave, density estimation, Kolmogorov-Smirnov test, \proglang{R}} \Plainkeywords{log-concave, density estimation, R} %% without formatting %% at least one keyword must be supplied %% publication information %% NOTE: Typically, this can be left commented and will be filled out by the technical editor \Volume{39} \Issue{6} \Month{March} \Year{2011} \Submitdate{2009-11-12} \Acceptdate{2010-10-06} %% The address of (at least) one author should be given %% in the following format: \Address{ Lutz D\"umbgen \\ Institute of Mathematical Statistics and Actuarial Science \\ University of Bern \\ 3012 Bern, Switzerland \\ Telephone: +41/0/31631-8802 \\ Fax: +41/0/31631-3805 \\ E-mail: \email{duembgen@stat.unibe.ch} \\ URL: \url{http://www.staff.unibe.ch/duembgen} \bigskip Kaspar Rufibach\\ Biostatistics Unit\\ Institute for Social and Preventive Medicine\\ University of Zurich\\ 8001 Zurich, Switzerland\\ Telephone: +41/0/44634-4643 \\ Fax: +41/0/44634-4386 \\ E-mail: \email{kaspar.rufibach@ifspm.uzh.ch}\\ URL: \url{http://www.biostat.uzh.ch/aboutus/people/rufibach.html} } %% It is also possible to add a telephone and fax number %% before the e-mail in the following format: %% Telephone: +43/1/31336-5053 %% Fax: +43/1/31336-734 %% for those who use Sweave please include the following line (with % symbols): %% need no \usepackage{Sweave.sty} %% end of declarations %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsthm} \usepackage{latexsym} \usepackage{thumbpdf} \newtheorem{theorem}{Theorem}[section] %\newtheorem{lemma}[theorem]{Lemma} %\newtheorem{corollary}[theorem]{Corollary} %\newtheorem{proposition}[theorem]{Proposition} %\newtheorem{example}[theorem]{Example} \newenvironment{Theorem}{\begin{theorem}\sl}{\end{theorem}} %\newenvironment{Lemma}{\begin{lemma}\sl}{\end{lemma}} %\newenvironment{Corollary}{\begin{corollary}\sl}{\end{corollary}} %\newenvironment{Proposition}{\begin{proposition}\sl}{\end{proposition}} %\newenvironment{Example}{\begin{example}\rm}{\end{example}} \def\bea{\begin{eqnarray*}} \def\eea{\end{eqnarray*}} \def\be{\begin{equation}} \def\ee{\end{equation}} \def\bean{\begin{eqnarray}} \def\eean{\end{eqnarray}} \def\barr{\begin{array}} \def\earr{\end{array}} \def\bdes{\begin{description}} \def\edes{\end{description}} \def\bi{\begin{itemize}} \def\ei{\end{itemize}} \def\hat{\widehat} \def\R{\mathbb{R}} \def\est{\widehat{f}} \def\lest{\widehat{\varphi}} \def\dest{\widehat{F}} \def\SS{\mathcal{S}} \newcommand{\ED}{\mathbb{F}} \def\d{\, \mathrm{d}} \def\Del{\Delta} \def\Bl{\Bigl} \def\Br{\Bigr} \newcommand{\ve}[1]{\mathbf{#1}} \def\Var{\mathop{\rm Var}\nolimits} \def\argmax{\mathop{\rm arg\,max}} \def\PP{{\cal P}} \def\II{{\cal I}} \def\KK{{\cal K}} \def\RR{{\cal R}} \newcommand{\SupN}[1]{\Vert #1 \Vert_\infty} \SweaveOpts{keep.source = TRUE} \SweaveOpts{prefix.string = plots/} <