mvtnorm/0000755000176200001440000000000014130012112011735 5ustar liggesusersmvtnorm/NAMESPACE0000644000176200001440000000050313525241121013165 0ustar liggesusers useDynLib(mvtnorm, .registration = TRUE) importFrom("stats", pnorm, qnorm, rnorm, runif, pt, qt, rchisq, uniroot, cov2cor, optim, coef, glm, pcauchy, qcauchy, predict, quasi) importFrom("methods", hasArg) export(rmvnorm, dmvnorm, pmvnorm, pmvt, rmvt, qmvnorm, qmvt, dmvt, GenzBretz, Miwa, TVPACK) mvtnorm/man/0000755000176200001440000000000014127061037012530 5ustar liggesusersmvtnorm/man/pmvnorm.Rd0000644000176200001440000001112114057355213014515 0ustar liggesusers\name{pmvnorm} \alias{pmvnorm} \title{ Multivariate Normal Distribution } \description{ Computes the distribution function of the multivariate normal distribution for arbitrary limits and correlation matrices. } \usage{ pmvnorm(lower=-Inf, upper=Inf, mean=rep(0, length(lower)), corr=NULL, sigma=NULL, algorithm = GenzBretz(), keepAttr=TRUE, ...) } \arguments{ \item{lower}{ the vector of lower limits of length n.} \item{upper}{ the vector of upper limits of length n.} \item{mean}{ the mean vector of length n.} \item{corr}{ the correlation matrix of dimension n.} \item{sigma}{ the covariance matrix of dimension n less than 1000. Either \code{corr} or \code{sigma} can be specified. If \code{sigma} is given, the problem is standardized. If neither \code{corr} nor \code{sigma} is given, the identity matrix is used for \code{sigma}. } \item{algorithm}{ an object of class \code{\link{GenzBretz}}, \code{\link{Miwa}} or \code{\link{TVPACK}} specifying both the algorithm to be used as well as the associated hyper parameters.} \item{keepAttr}{\code{\link{logical}} indicating if \code{\link{attributes}} such as \code{error} and \code{msg} should be attached to the return value. The default, \code{TRUE} is back compatible.} \item{...}{ additional parameters (currently given to \code{GenzBretz} for backward compatibility issues). } } \details{ This program involves the computation of multivariate normal probabilities with arbitrary correlation matrices. It involves both the computation of singular and nonsingular probabilities. The implemented methodology is described in Genz (1992, 1993) (for algorithm GenzBretz), in Miwa et al. (2003) for algorithm Miwa (useful up to dimension 20) and Genz (2004) for the TVPACK algorithm (which covers 2- and 3-dimensional problems for semi-infinite integration regions). Note the default algorithm GenzBretz is randomized and hence slightly depends on \code{\link{.Random.seed}} and that both \code{-Inf} and \code{+Inf} may be specified in \code{lower} and \code{upper}. For more details see \code{\link{pmvt}}. The multivariate normal case is treated as a special case of \code{\link{pmvt}} with \code{df=0} and univariate problems are passed to \code{\link{pnorm}}. The multivariate normal density and random deviates are available using \code{\link{dmvnorm}} and \code{\link{rmvnorm}}. } \value{ The evaluated distribution function is returned, if \code{keepAttr} is true, with attributes \item{error}{estimated absolute error} \item{msg}{status message(s).} \item{algorithm}{a \code{\link{character}} string with \code{class(algorithm)}.} } \references{ Genz, A. (1992). Numerical computation of multivariate normal probabilities. \emph{Journal of Computational and Graphical Statistics}, \bold{1}, 141--150. Genz, A. (1993). Comparison of methods for the computation of multivariate normal probabilities. \emph{Computing Science and Statistics}, \bold{25}, 400--405. Genz, A. (2004), Numerical computation of rectangular bivariate and trivariate normal and t-probabilities, \emph{Statistics and Computing}, \bold{14}, 251--260. Genz, A. and Bretz, F. (2009), \emph{Computation of Multivariate Normal and t Probabilities}. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg. Miwa, A., Hayter J. and Kuriki, S. (2003). The evaluation of general non-centred orthant probabilities. \emph{Journal of the Royal Statistical Society}, Ser. B, 65, 223--234. } \source{ \url{http://www.sci.wsu.edu/math/faculty/genz/homepage} } \seealso{\code{\link{qmvnorm}}} \examples{ n <- 5 mean <- rep(0, 5) lower <- rep(-1, 5) upper <- rep(3, 5) corr <- diag(5) corr[lower.tri(corr)] <- 0.5 corr[upper.tri(corr)] <- 0.5 prob <- pmvnorm(lower, upper, mean, corr) print(prob) stopifnot(pmvnorm(lower=-Inf, upper=3, mean=0, sigma=1) == pnorm(3)) a <- pmvnorm(lower=-Inf,upper=c(.3,.5),mean=c(2,4),diag(2)) stopifnot(round(a,16) == round(prod(pnorm(c(.3,.5),c(2,4))),16)) a <- pmvnorm(lower=-Inf,upper=c(.3,.5,1),mean=c(2,4,1),diag(3)) stopifnot(round(a,16) == round(prod(pnorm(c(.3,.5,1),c(2,4,1))),16)) # Example from R News paper (original by Genz, 1992): m <- 3 sigma <- diag(3) sigma[2,1] <- 3/5 sigma[3,1] <- 1/3 sigma[3,2] <- 11/15 pmvnorm(lower=rep(-Inf, m), upper=c(1,4,2), mean=rep(0, m), corr=sigma) # Correlation and Covariance a <- pmvnorm(lower=-Inf, upper=c(2,2), sigma = diag(2)*2) b <- pmvnorm(lower=-Inf, upper=c(2,2)/sqrt(2), corr=diag(2)) stopifnot(all.equal(round(a,5) , round(b, 5))) } \keyword{distribution} mvtnorm/man/algorithms.Rd0000644000176200001440000000653114113171051015166 0ustar liggesusers\name{algorithms} \alias{GenzBretz} \alias{Miwa} \alias{TVPACK} \title{ Choice of Algorithm and Hyper Parameters } \description{ Choose between three algorithms for evaluating normal (and t-) distributions and define hyper parameters. } \usage{ GenzBretz(maxpts = 25000, abseps = 0.001, releps = 0) Miwa(steps = 128, checkCorr = TRUE, maxval = 1e3) TVPACK(abseps = 1e-6) } \arguments{ \item{maxpts}{maximum number of function values as integer. The internal FORTRAN code always uses a minimum number depending on the dimension. (for example 752 for three-dimensional problems).} \item{abseps}{absolute error tolerance; for \code{TVPACK} only used for dimension 3.} \item{releps}{ relative error tolerance as double. } \item{steps}{number of grid points to be evaluated; cannot be larger than 4097.} \item{checkCorr}{logical indicating if a check for singularity of the correlation matrix should be performed (once per function call to \code{pmvt()} or \code{pmvnorm()}).} \item{maxval}{replacement for \code{Inf} when non-orthant probabilities involving \code{Inf} shall be computed.} } \details{ There are three algorithms available for evaluating normal (and two algorithms for t-) probabilities: The default is the randomized Quasi-Monte-Carlo procedure by Genz (1992, 1993) and Genz and Bretz (2002) applicable to arbitrary covariance structures and dimensions up to 1000. For normal probabilities, smaller dimensions (up to 20) and non-singular covariance matrices, the algorithm by Miwa et al. (2003) can be used as well. This algorithm can compute orthant probabilities (\code{lower} being \code{-Inf} or \code{upper} equal to \code{Inf}). Non-orthant probabilities are computed from the corresponding orthant probabilities, however, infinite limits are replaced by \code{maxval} along with a warning. For two- and three-dimensional problems and semi-infinite integration region, \code{TVPACK} implements an interface to the methods described by Genz (2004). } \value{ An object of class \code{"GenzBretz"}, \code{"Miwa"}, or \code{"TVPACK"} defining hyper parameters. } \references{ Genz, A. (1992). Numerical computation of multivariate normal probabilities. \emph{Journal of Computational and Graphical Statistics}, \bold{1}, 141--150. Genz, A. (1993). Comparison of methods for the computation of multivariate normal probabilities. \emph{Computing Science and Statistics}, \bold{25}, 400--405. Genz, A. and Bretz, F. (2002), Methods for the computation of multivariate t-probabilities. \emph{Journal of Computational and Graphical Statistics}, \bold{11}, 950--971. Genz, A. (2004), Numerical computation of rectangular bivariate and trivariate normal and t-probabilities, \emph{Statistics and Computing}, \bold{14}, 251--260. Genz, A. and Bretz, F. (2009), \emph{Computation of Multivariate Normal and t Probabilities}. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg. Miwa, A., Hayter J. and Kuriki, S. (2003). The evaluation of general non-centred orthant probabilities. \emph{Journal of the Royal Statistical Society}, Ser. B, 65, 223--234. Mi, X., Miwa, T. and Hothorn, T. (2009). \code{mvtnorm}: New numerical algorithm for multivariate normal probabilities. \emph{The R Journal} \bold{1}(1): 37--39. \url{https://journal.r-project.org/archive/2009-1/RJournal_2009-1_Mi+et+al.pdf} } \keyword{distribution} mvtnorm/man/qmvnorm.Rd0000644000176200001440000000572713525241121014524 0ustar liggesusers\name{qmvnorm} \alias{qmvnorm} \title{ Quantiles of the Multivariate Normal Distribution } \description{ Computes the equicoordinate quantile function of the multivariate normal distribution for arbitrary correlation matrices based on inversion of \code{\link{pmvnorm}}, using a stochastic root finding algorithm described in Bornkamp (2018). } \usage{ qmvnorm(p, interval = NULL, tail = c("lower.tail", "upper.tail", "both.tails"), mean = 0, corr = NULL, sigma = NULL, algorithm = GenzBretz(), ptol = 0.001, maxiter = 500, trace = FALSE, ...) } %- maybe also 'usage' for other objects documented here. \arguments{ \item{p}{ probability.} \item{interval}{ optional, a vector containing the end-points of the interval to be searched. Does not need to contain the true quantile, just used as starting values by the root-finder. If equal to NULL a guess is used.} \item{tail}{ specifies which quantiles should be computed. \code{lower.tail} gives the quantile \eqn{x} for which \eqn{P[X \le x] = p}, \code{upper.tail} gives \eqn{x} with \eqn{P[X > x] = p} and \code{both.tails} leads to \eqn{x} with \eqn{P[-x \le X \le x] = p}.} \item{mean}{ the mean vector of length n. } \item{corr}{ the correlation matrix of dimension n.} \item{sigma}{ the covariance matrix of dimension n. Either \code{corr} or \code{sigma} can be specified. If \code{sigma} is given, the problem is standardized. If neither \code{corr} nor \code{sigma} is given, the identity matrix is used for \code{sigma}. } \item{algorithm}{ an object of class \code{\link{GenzBretz}}, \code{\link{Miwa}} or \code{\link{TVPACK}} specifying both the algorithm to be used as well as the associated hyper parameters.} \item{ptol, maxiter, trace}{Parameters passed to the stochastic root-finding algorithm. Iteration stops when the 95\% confidence interval for the predicted quantile is inside [p-ptol, p+ptol]. \code{maxiter} is the maximum number of iterations for the root finding algorithm. \code{trace} prints the iterations of the root finder.} \item{...}{ additional parameters to be passed to \code{\link{GenzBretz}}.} } \details{ Only equicoordinate quantiles are computed, i.e., the quantiles in each dimension coincide. The result is seed dependend. } \value{ A list with two components: \code{quantile} and \code{f.quantile} give the location of the quantile and the difference between the distribution function evaluated at the quantile and \code{p}. } \references{ Bornkamp, B. (2018). Calculating quantiles of noisy distribution functions using local linear regressions. \emph{Computational Statistics}, \bold{33}, 487--501. } \seealso{\code{\link{pmvnorm}}, \code{\link{qmvt}}} \examples{ qmvnorm(0.95, sigma = diag(2), tail = "both") } \keyword{distribution} mvtnorm/man/pmvt.Rd0000644000176200001440000001653514057355213014023 0ustar liggesusers\name{pmvt} \alias{pmvt} \title{ Multivariate t Distribution } \description{ Computes the the distribution function of the multivariate t distribution for arbitrary limits, degrees of freedom and correlation matrices based on algorithms by Genz and Bretz. } \usage{ pmvt(lower=-Inf, upper=Inf, delta=rep(0, length(lower)), df=1, corr=NULL, sigma=NULL, algorithm = GenzBretz(), type = c("Kshirsagar", "shifted"), keepAttr=TRUE, ...) } \arguments{ \item{lower}{ the vector of lower limits of length n.} \item{upper}{ the vector of upper limits of length n.} \item{delta}{ the vector of noncentrality parameters of length n, for \code{type = "shifted"} delta specifies the mode.} \item{df}{ degree of freedom as integer. Normal probabilities are computed for \code{df=0}.} \item{corr}{ the correlation matrix of dimension n.} \item{sigma}{ the scale matrix of dimension n. Either \code{corr} or \code{sigma} can be specified. If \code{sigma} is given, the problem is standardized. If neither \code{corr} nor \code{sigma} is given, the identity matrix is used for \code{sigma}. } \item{algorithm}{ an object of class \code{\link{GenzBretz}} or \code{\link{TVPACK}} defining the hyper parameters of this algorithm.} \item{type}{ type of the noncentral multivariate t distribution to be computed. \code{type = "Kshirsagar"} corresponds to formula (1.4) in Genz and Bretz (2009) (see also Chapter 5.1 in Kotz and Nadarajah (2004)). This is the noncentral t-distribution needed for calculating the power of multiple contrast tests under a normality assumption. \code{type = "shifted"} corresponds to the formula right before formula (1.4) in Genz and Bretz (2009) (see also formula (1.1) in Kotz and Nadarajah (2004)). It is a location shifted version of the central t-distribution. This noncentral multivariate t distribution appears for example as the Bayesian posterior distribution for the regression coefficients in a linear regression. In the central case both types coincide. } \item{keepAttr}{\code{\link{logical}} indicating if \code{\link{attributes}} such as \code{error} and \code{msg} should be attached to the return value. The default, \code{TRUE} is back compatible.} \item{...}{additional parameters (currently given to \code{GenzBretz} for backward compatibility issues). } } \details{ This function involves the computation of central and noncentral multivariate t-probabilities with arbitrary correlation matrices. It involves both the computation of singular and nonsingular probabilities. The methodology (for default \code{algorithm = GenzBretz()}) is based on randomized quasi Monte Carlo methods and described in Genz and Bretz (1999, 2002). \cr Because of the randomization, the result for this algorithm (slightly) depends on \code{\link{.Random.seed}}. For 2- and 3-dimensional problems one can also use the \code{\link{TVPACK}} routines described by Genz (2004), which only handles semi-infinite integration regions (and for \code{type = "Kshirsagar"} only central problems). For \code{type = "Kshirsagar"} and a given correlation matrix \code{corr}, for short \eqn{A}, say, (which has to be positive semi-definite) and degrees of freedom \eqn{\nu} the following values are numerically evaluated %% FIXME add non-LaTeX alternative \deqn{LaTeX}{non-LaTex} \deqn{I = 2^{1-\nu/2} / \Gamma(\nu/2) \int_0^\infty s^{\nu-1} \exp(-s^2/2) \Phi(s \cdot lower/\sqrt{\nu} - \delta, s \cdot upper/\sqrt{\nu} - \delta) \, ds } where \deqn{\Phi(a,b) = (det(A)(2\pi)^m)^{-1/2} \int_a^b \exp(-x^\prime Ax/2) \, dx} is the multivariate normal distribution and \eqn{m} is the number of rows of \eqn{A}. For \code{type = "shifted"}, a positive definite symmetric matrix \eqn{S} (which might be the correlation or the scale matrix), mode (vector) \eqn{\delta} and degrees of freedom \eqn{\nu} the following integral is evaluated: \deqn{c\int_{lower_1}^{upper_1}...\int_{lower_m}^{upper_m} (1+(x-\delta)'S^{-1}(x-\delta)/\nu)^{-(\nu+m)/2}\, dx_1 ... dx_m, } where \deqn{c = \Gamma((\nu+m)/2)/((\pi \nu)^{m/2}\Gamma(\nu/2)|S|^{1/2}),} and \eqn{m} is the number of rows of \eqn{S}. Note that both \code{-Inf} and \code{+Inf} may be specified in the lower and upper integral limits in order to compute one-sided probabilities. Univariate problems are passed to \code{\link{pt}}. If \code{df = 0}, normal probabilities are returned. } \value{ The evaluated distribution function is returned, if \code{keepAttr} is true, with attributes \item{error}{estimated absolute error and} \item{msg}{status message (a \code{\link{character}} string).} \item{algorithm}{a \code{\link{character}} string with \code{class(algorithm)}.} } \references{ Genz, A. and Bretz, F. (1999), Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. \emph{Journal of Statistical Computation and Simulation}, \bold{63}, 361--378. Genz, A. and Bretz, F. (2002), Methods for the computation of multivariate t-probabilities. \emph{Journal of Computational and Graphical Statistics}, \bold{11}, 950--971. Genz, A. (2004), Numerical computation of rectangular bivariate and trivariate normal and t-probabilities, \emph{Statistics and Computing}, \bold{14}, 251--260. Genz, A. and Bretz, F. (2009), \emph{Computation of Multivariate Normal and t Probabilities}. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg. S. Kotz and S. Nadarajah (2004), \emph{Multivariate t Distributions and Their Applications}. Cambridge University Press. Cambridge. Edwards D. and Berry, Jack J. (1987), The efficiency of simulation-based multiple comparisons. \emph{Biometrics}, \bold{43}, 913--928. } \source{ \url{http://www.sci.wsu.edu/math/faculty/genz/homepage} } \seealso{\code{\link{qmvt}}} \examples{ n <- 5 lower <- -1 upper <- 3 df <- 4 corr <- diag(5) corr[lower.tri(corr)] <- 0.5 delta <- rep(0, 5) prob <- pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=corr) print(prob) pmvt(lower=-Inf, upper=3, df = 3, sigma = 1) == pt(3, 3) # Example from R News paper (original by Edwards and Berry, 1987) n <- c(26, 24, 20, 33, 32) V <- diag(1/n) df <- 130 C <- c(1,1,1,0,0,-1,0,0,1,0,0,-1,0,0,1,0,0,0,-1,-1,0,0,-1,0,0) C <- matrix(C, ncol=5) ### scale matrix cv <- C \%*\% V \%*\% t(C) ### correlation matrix dv <- t(1/sqrt(diag(cv))) cr <- cv * (t(dv) \%*\% dv) delta <- rep(0,5) myfct <- function(q, alpha) { lower <- rep(-q, ncol(cv)) upper <- rep(q, ncol(cv)) pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=cr, abseps=0.0001) - alpha } ### uniroot for this simple problem round(uniroot(myfct, lower=1, upper=5, alpha=0.95)$root, 3) # compare pmvt and pmvnorm for large df: a <- pmvnorm(lower=-Inf, upper=1, mean=rep(0, 5), corr=diag(5)) b <- pmvt(lower=-Inf, upper=1, delta=rep(0, 5), df=300, corr=diag(5)) a b stopifnot(round(a, 2) == round(b, 2)) # correlation and scale matrix a <- pmvt(lower=-Inf, upper=2, delta=rep(0,5), df=3, sigma = diag(5)*2) b <- pmvt(lower=-Inf, upper=2/sqrt(2), delta=rep(0,5), df=3, corr=diag(5)) attributes(a) <- NULL attributes(b) <- NULL a b stopifnot(all.equal(round(a,3) , round(b, 3))) a <- pmvt(0, 1,df=10) attributes(a) <- NULL b <- pt(1, df=10) - pt(0, df=10) stopifnot(all.equal(round(a,10) , round(b, 10))) } \keyword{distribution} mvtnorm/man/Mvnorm.Rd0000644000176200001440000000440013667705060014304 0ustar liggesusers\name{Mvnorm} \alias{dmvnorm} \alias{rmvnorm} \title{Multivariate Normal Density and Random Deviates} \description{ These functions provide the density function and a random number generator for the multivariate normal distribution with mean equal to \code{mean} and covariance matrix \code{sigma}. } \usage{ dmvnorm(x, mean = rep(0, p), sigma = diag(p), log = FALSE, checkSymmetry = TRUE) rmvnorm(n, mean = rep(0, nrow(sigma)), sigma = diag(length(mean)), method=c("eigen", "svd", "chol"), pre0.9_9994 = FALSE, checkSymmetry = TRUE) } \arguments{ \item{x}{vector or matrix of quantiles. If \code{x} is a matrix, each row is taken to be a quantile.} \item{n}{number of observations.} \item{mean}{mean vector, default is \code{rep(0, length = ncol(x))}.} \item{sigma}{covariance matrix, default is \code{diag(ncol(x))}.} \item{log}{logical; if \code{TRUE}, densities d are given as log(d).} \item{method}{string specifying the matrix decomposition used to determine the matrix root of \code{sigma}. Possible methods are eigenvalue decomposition (\code{"eigen"}, default), singular value decomposition (\code{"svd"}), and Cholesky decomposition (\code{"chol"}). The Cholesky is typically fastest, not by much though.} \item{pre0.9_9994}{logical; if \code{FALSE}, the output produced in mvtnorm versions up to 0.9-9993 is reproduced. In 0.9-9994, the output is organized such that \code{rmvnorm(10,...)} has the same first ten rows as \code{rmvnorm(100, ...)} when called with the same seed.} \item{checkSymmetry}{logical; if \code{FALSE}, skip checking whether the covariance matrix is symmetric or not. This will speed up the computation but may cause unexpected outputs when ill-behaved \code{sigma} is provided. The default value is \code{TRUE}.} } \author{Friedrich Leisch and Fabian Scheipl} \seealso{\code{\link{pmvnorm}}, \code{\link{rnorm}}, \code{\link{qmvnorm}}} \examples{ dmvnorm(x=c(0,0)) dmvnorm(x=c(0,0), mean=c(1,1)) sigma <- matrix(c(4,2,2,3), ncol=2) x <- rmvnorm(n=500, mean=c(1,2), sigma=sigma) colMeans(x) var(x) x <- rmvnorm(n=500, mean=c(1,2), sigma=sigma, method="chol") colMeans(x) var(x) plot(x) } \keyword{distribution} \keyword{multivariate} mvtnorm/man/qmvt.Rd0000644000176200001440000001103613525241121014002 0ustar liggesusers\name{qmvt} \alias{qmvt} \title{ Quantiles of the Multivariate t Distribution } \description{ Computes the equicoordinate quantile function of the multivariate t distribution for arbitrary correlation matrices based on inversion of \code{\link{pmvt}}, using a stochastic root finding algorithm described in Bornkamp (2018). } \usage{ qmvt(p, interval = NULL, tail = c("lower.tail", "upper.tail", "both.tails"), df = 1, delta = 0, corr = NULL, sigma = NULL, algorithm = GenzBretz(), type = c("Kshirsagar", "shifted"), ptol = 0.001, maxiter = 500, trace = FALSE, ...) } \arguments{ \item{p}{ probability.} \item{interval}{ optional, a vector containing the end-points of the interval to be searched. Does not need to contain the true quantile, just used as starting values by the root-finder. If equal to NULL a guess is used.} \item{tail}{ specifies which quantiles should be computed. \code{lower.tail} gives the quantile \eqn{x} for which \eqn{P[X \le x] = p}, \code{upper.tail} gives \eqn{x} with \eqn{P[X > x] = p} and \code{both.tails} leads to \eqn{x} with \eqn{P[-x \le X \le x] = p}.} \item{delta}{ the vector of noncentrality parameters of length n, for \code{type = "shifted"} delta specifies the mode.} \item{df}{ degree of freedom as integer. Normal quantiles are computed for \code{df = 0} or \code{df = Inf}.} \item{corr}{ the correlation matrix of dimension n.} \item{sigma}{ the covariance matrix of dimension n. Either \code{corr} or \code{sigma} can be specified. If \code{sigma} is given, the problem is standardized. If neither \code{corr} nor \code{sigma} is given, the identity matrix in the univariate case (so \code{corr = 1}) is used for \code{corr}. } \item{algorithm}{ an object of class \code{\link{GenzBretz}} or \code{\link{TVPACK}} defining the hyper parameters of this algorithm.} \item{type}{ type of the noncentral multivariate t distribution to be computed. \code{type = "Kshirsagar"} corresponds to formula (1.4) in Genz and Bretz (2009) (see also Chapter 5.1 in Kotz and Nadarajah (2004)) and \code{type = "shifted"} corresponds to the formula before formula (1.4) in Genz and Bretz (2009) (see also formula (1.1) in Kotz and Nadarajah (2004)). } \item{ptol, maxiter, trace}{Parameters passed to the stochastic root-finding algorithm. Iteration stops when the 95\% confidence interval for the predicted quantile is inside [p-ptol, p+ptol]. \code{maxiter} is the maximum number of iterations for the root finding algorithm. \code{trace} prints the iterations of the root finder.} \item{...}{ additional parameters to be passed to \code{\link{GenzBretz}}.} } \details{ Only equicoordinate quantiles are computed, i.e., the quantiles in each dimension coincide. The result is seed dependend. } \value{ A list with two components: \code{quantile} and \code{f.quantile} give the location of the quantile and the difference between the distribution function evaluated at the quantile and \code{p}. } \references{ Bornkamp, B. (2018). Calculating quantiles of noisy distribution functions using local linear regressions. \emph{Computational Statistics}, \bold{33}, 487--501. } \seealso{\code{\link{pmvnorm}}, \code{\link{qmvnorm}}} \examples{ ## basic evaluation qmvt(0.95, df = 16, tail = "both") ## check behavior for df=0 and df=Inf Sigma <- diag(2) set.seed(29) q0 <- qmvt(0.95, sigma = Sigma, df = 0, tail = "both")$quantile set.seed(29) q8 <- qmvt(0.95, sigma = Sigma, df = Inf, tail = "both")$quantile set.seed(29) qn <- qmvnorm(0.95, sigma = Sigma, tail = "both")$quantile stopifnot(identical(q0, q8), isTRUE(all.equal(q0, qn, tol = (.Machine$double.eps)^(1/3)))) ## if neither sigma nor corr are provided, corr = 1 is used internally df <- 0 set.seed(29) qt95 <- qmvt(0.95, df = df, tail = "both")$quantile set.seed(29) qt95.c <- qmvt(0.95, df = df, corr = 1, tail = "both")$quantile set.seed(29) qt95.s <- qmvt(0.95, df = df, sigma = 1, tail = "both")$quantile stopifnot(identical(qt95, qt95.c), identical(qt95, qt95.s)) df <- 4 set.seed(29) qt95 <- qmvt(0.95, df = df, tail = "both")$quantile set.seed(29) qt95.c <- qmvt(0.95, df = df, corr = 1, tail = "both")$quantile set.seed(29) qt95.s <- qmvt(0.95, df = df, sigma = 1, tail = "both")$quantile stopifnot(identical(qt95, qt95.c), identical(qt95, qt95.s)) } \keyword{distribution} mvtnorm/man/Mvt.Rd0000644000176200001440000001250213667705060013576 0ustar liggesusers\name{Mvt} \alias{dmvt} \alias{rmvt} \title{The Multivariate t Distribution} \description{ These functions provide information about the multivariate \eqn{t} distribution with non-centrality parameter (or mode) \code{delta}, scale matrix \code{sigma} and degrees of freedom \code{df}. \code{dmvt} gives the density and \code{rmvt} generates random deviates. } \usage{ rmvt(n, sigma = diag(2), df = 1, delta = rep(0, nrow(sigma)), type = c("shifted", "Kshirsagar"), ...) dmvt(x, delta = rep(0, p), sigma = diag(p), df = 1, log = TRUE, type = "shifted", checkSymmetry = TRUE) } \arguments{ \item{x}{vector or matrix of quantiles. If \code{x} is a matrix, each row is taken to be a quantile.} \item{n}{number of observations.} \item{delta}{the vector of noncentrality parameters of length n, for \code{type = "shifted"} delta specifies the mode.} \item{sigma}{scale matrix, defaults to \code{diag(ncol(x))}.} \item{df}{degrees of freedom. \code{df = 0} or \code{df = Inf} corresponds to the multivariate normal distribution.} \item{log}{\code{\link{logical}} indicating whether densities \eqn{d} are given as \eqn{\log(d)}{log(d)}.} \item{type}{type of the noncentral multivariate \eqn{t} distribution. \code{type = "Kshirsagar"} corresponds to formula (1.4) in Genz and Bretz (2009) (see also Chapter 5.1 in Kotz and Nadarajah (2004)). This is the noncentral t-distribution needed for calculating the power of multiple contrast tests under a normality assumption. \code{type = "shifted"} corresponds to the formula right before formula (1.4) in Genz and Bretz (2009) (see also formula (1.1) in Kotz and Nadarajah (2004)). It is a location shifted version of the central t-distribution. This noncentral multivariate \eqn{t} distribution appears for example as the Bayesian posterior distribution for the regression coefficients in a linear regression. In the central case both types coincide. Note that the defaults differ from the default in \code{\link{pmvt}()} (for reasons of backward compatibility).} \item{checkSymmetry}{logical; if \code{FALSE}, skip checking whether the covariance matrix is symmetric or not. This will speed up the computation but may cause unexpected outputs when ill-behaved \code{sigma} is provided. The default value is \code{TRUE}.} \item{\dots}{additional arguments to \code{\link{rmvnorm}()}, for example \code{method}.} } \details{ If \eqn{\bm{X}}{X} denotes a random vector following a \eqn{t} distribution with location vector \eqn{\bm{0}}{0} and scale matrix \eqn{\Sigma}{Sigma} (written \eqn{X\sim t_\nu(\bm{0},\Sigma)}{X ~ t_nu(0, Sigma)}), the scale matrix (the argument \code{sigma}) is not equal to the covariance matrix \eqn{Cov(\bm{X})}{Cov(X)} of \eqn{\bm{X}}{X}. If the degrees of freedom \eqn{\nu}{nu} (the argument \code{df}) is larger than 2, then \eqn{Cov(\bm{X})=\Sigma\nu/(\nu-2)}{Cov(X)=Sigma nu/(nu-2)}. Furthermore, in this case the correlation matrix \eqn{Cor(\bm{X})}{Cor(X)} equals the correlation matrix corresponding to the scale matrix \eqn{\Sigma}{Sigma} (which can be computed with \code{\link{cov2cor}()}). Note that the scale matrix is sometimes referred to as \dQuote{dispersion matrix}; see McNeil, Frey, Embrechts (2005, p. 74). For \code{type = "shifted"} the density \deqn{c(1+(x-\delta)'S^{-1}(x-\delta)/\nu)^{-(\nu+m)/2}} is implemented, where \deqn{c = \Gamma((\nu+m)/2)/((\pi \nu)^{m/2}\Gamma(\nu/2)|S|^{1/2}),} \eqn{S} is a positive definite symmetric matrix (the matrix \code{sigma} above), \eqn{\delta}{delta} is the non-centrality vector and \eqn{\nu}{nu} are the degrees of freedom. \code{df=0} historically leads to the multivariate normal distribution. From a mathematical point of view, rather \code{df=Inf} corresponds to the multivariate normal distribution. This is (now) also allowed for \code{rmvt()} and \code{dmvt()}. Note that \code{dmvt()} has default \code{log = TRUE}, whereas \code{\link{dmvnorm}()} has default \code{log = FALSE}. } \references{ McNeil, A. J., Frey, R., and Embrechts, P. (2005). \emph{Quantitative Risk Management: Concepts, Techniques, Tools}. Princeton University Press. } \seealso{\code{\link{pmvt}()} and \code{\link{qmvt}()}} \examples{ ## basic evaluation dmvt(x = c(0,0), sigma = diag(2)) ## check behavior for df=0 and df=Inf x <- c(1.23, 4.56) mu <- 1:2 Sigma <- diag(2) x0 <- dmvt(x, delta = mu, sigma = Sigma, df = 0) # default log = TRUE! x8 <- dmvt(x, delta = mu, sigma = Sigma, df = Inf) # default log = TRUE! xn <- dmvnorm(x, mean = mu, sigma = Sigma, log = TRUE) stopifnot(identical(x0, x8), identical(x0, xn)) ## X ~ t_3(0, diag(2)) x <- rmvt(100, sigma = diag(2), df = 3) # t_3(0, diag(2)) sample plot(x) ## X ~ t_3(mu, Sigma) n <- 1000 mu <- 1:2 Sigma <- matrix(c(4, 2, 2, 3), ncol=2) set.seed(271) x <- rep(mu, each=n) + rmvt(n, sigma=Sigma, df=3) plot(x) ## Note that the call rmvt(n, mean=mu, sigma=Sigma, df=3) does *not* ## give a valid sample from t_3(mu, Sigma)! [and thus throws an error] try(rmvt(n, mean=mu, sigma=Sigma, df=3)) ## df=Inf correctly samples from a multivariate normal distribution set.seed(271) x <- rep(mu, each=n) + rmvt(n, sigma=Sigma, df=Inf) set.seed(271) x. <- rmvnorm(n, mean=mu, sigma=Sigma) stopifnot(identical(x, x.)) } \keyword{distribution} \keyword{multivariate} mvtnorm/DESCRIPTION0000644000176200001440000000277414130012112013455 0ustar liggesusersPackage: mvtnorm Title: Multivariate Normal and t Distributions Version: 1.1-3 Date: 2021-10-05 Authors@R: c(person("Alan", "Genz", role = "aut"), person("Frank", "Bretz", role = "aut"), person("Tetsuhisa", "Miwa", role = "aut"), person("Xuefei", "Mi", role = "aut"), person("Friedrich", "Leisch", role = "ctb"), person("Fabian", "Scheipl", role = "ctb"), person("Bjoern", "Bornkamp", role = "ctb", comment = c(ORCID = "0000-0002-6294-8185")), person("Martin", "Maechler", role = "ctb", comment = c(ORCID = "0000-0002-8685-9910")), person("Torsten", "Hothorn", role = c("aut", "cre"), email = "Torsten.Hothorn@R-project.org", comment = c(ORCID = "0000-0001-8301-0471"))) Description: Computes multivariate normal and t probabilities, quantiles, random deviates and densities. Imports: stats, methods Depends: R(>= 3.5.0) License: GPL-2 URL: http://mvtnorm.R-forge.R-project.org NeedsCompilation: yes Packaged: 2021-10-05 14:35:36 UTC; hothorn Author: Alan Genz [aut], Frank Bretz [aut], Tetsuhisa Miwa [aut], Xuefei Mi [aut], Friedrich Leisch [ctb], Fabian Scheipl [ctb], Bjoern Bornkamp [ctb] (), Martin Maechler [ctb] (), Torsten Hothorn [aut, cre] () Maintainer: Torsten Hothorn Repository: CRAN Date/Publication: 2021-10-08 09:50:02 UTC mvtnorm/build/0000755000176200001440000000000014127061270013053 5ustar liggesusersmvtnorm/build/vignette.rds0000644000176200001440000000031414127061270015410 0ustar liggesusersb```b`add`b2 1# ' K-/ +G -KW-+/ŭ % MX#R%@„5/17M?KjAj^ HvѴpxVaaqIY0AAn0Ez0?Ht&${+%$Q/n.mvtnorm/tests/0000755000176200001440000000000014127061037013117 5ustar liggesusersmvtnorm/tests/Examples/0000755000176200001440000000000013525241121014670 5ustar liggesusersmvtnorm/tests/Examples/mvtnorma-Ex.Rout.save0000644000176200001440000002656413525241121020732 0ustar liggesusers R version 3.2.3 (2015-12-10) -- "Wooden Christmas-Tree" Copyright (C) 2015 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type 'license()' or 'licence()' for distribution details. Natural language support but running in an English locale R is a collaborative project with many contributors. Type 'contributors()' for more information and 'citation()' on how to cite R or R packages in publications. Type 'demo()' for some demos, 'help()' for on-line help, or 'help.start()' for an HTML browser interface to help. Type 'q()' to quit R. > pkgname <- "mvtnorm" > source(file.path(R.home("share"), "R", "examples-header.R")) > options(warn = 1) > base::assign(".ExTimings", "mvtnorm-Ex.timings", pos = 'CheckExEnv') > base::cat("name\tuser\tsystem\telapsed\n", file=base::get(".ExTimings", pos = 'CheckExEnv')) > base::assign(".format_ptime", + function(x) { + if(!is.na(x[4L])) x[1L] <- x[1L] + x[4L] + if(!is.na(x[5L])) x[2L] <- x[2L] + x[5L] + options(OutDec = '.') + format(x[1L:3L], digits = 7L) + }, + pos = 'CheckExEnv') > > ### * > library('mvtnorm') > > base::assign(".oldSearch", base::search(), pos = 'CheckExEnv') > cleanEx() > nameEx("Mvnorm") > ### * Mvnorm > > flush(stderr()); flush(stdout()) > > base::assign(".ptime", proc.time(), pos = "CheckExEnv") > ### Name: Mvnorm > ### Title: Multivariate Normal Density and Random Deviates > ### Aliases: dmvnorm rmvnorm > ### Keywords: distribution multivariate > > ### ** Examples > > dmvnorm(x=c(0,0)) [1] 0.1591549 > dmvnorm(x=c(0,0), mean=c(1,1)) [1] 0.05854983 > > sigma <- matrix(c(4,2,2,3), ncol=2) > x <- rmvnorm(n=500, mean=c(1,2), sigma=sigma) > colMeans(x) [1] 0.9492868 1.9916602 > var(x) [,1] [,2] [1,] 4.276371 2.105874 [2,] 2.105874 3.186058 > > x <- rmvnorm(n=500, mean=c(1,2), sigma=sigma, method="chol") > colMeans(x) [1] 0.8724731 1.9804160 > var(x) [,1] [,2] [1,] 4.804847 2.430442 [2,] 2.430442 3.148344 > > plot(x) > > > > base::assign(".dptime", (proc.time() - get(".ptime", pos = "CheckExEnv")), pos = "CheckExEnv") > base::cat("Mvnorm", base::get(".format_ptime", pos = 'CheckExEnv')(get(".dptime", pos = "CheckExEnv")), "\n", file=base::get(".ExTimings", pos = 'CheckExEnv'), append=TRUE, sep="\t") > cleanEx() > nameEx("Mvt") > ### * Mvt > > flush(stderr()); flush(stdout()) > > base::assign(".ptime", proc.time(), pos = "CheckExEnv") > ### Name: Mvt > ### Title: The Multivariate t Distribution > ### Aliases: dmvt rmvt > ### Keywords: distribution multivariate > > ### ** Examples > > ## basic evaluation > dmvt(x = c(0,0), sigma = diag(2)) [1] -1.837877 > > ## check behavior for df=0 and df=Inf > x <- c(1.23, 4.56) > mu <- 1:2 > Sigma <- diag(2) > x0 <- dmvt(x, delta = mu, sigma = Sigma, df = 0) # default log = TRUE! > x8 <- dmvt(x, delta = mu, sigma = Sigma, df = Inf) # default log = TRUE! > xn <- dmvnorm(x, mean = mu, sigma = Sigma, log = TRUE) > stopifnot(identical(x0, x8), identical(x0, xn)) > > ## X ~ t_3(0, diag(2)) > x <- rmvt(100, sigma = diag(2), df = 3) # t_3(0, diag(2)) sample > plot(x) > > ## X ~ t_3(mu, Sigma) > n <- 1000 > mu <- 1:2 > Sigma <- matrix(c(4, 2, 2, 3), ncol=2) > set.seed(271) > x <- rep(mu, each=n) + rmvt(n, sigma=Sigma, df=3) > plot(x) > > ## Note that the call rmvt(n, mean=mu, sigma=Sigma, df=3) does *not* > ## give a valid sample from t_3(mu, Sigma)! [and thus throws an error] > try(rmvt(n, mean=mu, sigma=Sigma, df=3)) Error in rmvt(n, mean = mu, sigma = Sigma, df = 3) : Providing 'mean' does *not* sample from a multivariate t distribution! > > ## df=Inf correctly samples from a multivariate normal distribution > set.seed(271) > x <- rep(mu, each=n) + rmvt(n, sigma=Sigma, df=Inf) > set.seed(271) > x. <- rmvnorm(n, mean=mu, sigma=Sigma) > stopifnot(identical(x, x.)) > > > > base::assign(".dptime", (proc.time() - get(".ptime", pos = "CheckExEnv")), pos = "CheckExEnv") > base::cat("Mvt", base::get(".format_ptime", pos = 'CheckExEnv')(get(".dptime", pos = "CheckExEnv")), "\n", file=base::get(".ExTimings", pos = 'CheckExEnv'), append=TRUE, sep="\t") > cleanEx() > nameEx("pmvnorm") > ### * pmvnorm > > flush(stderr()); flush(stdout()) > > base::assign(".ptime", proc.time(), pos = "CheckExEnv") > ### Name: pmvnorm > ### Title: Multivariate Normal Distribution > ### Aliases: pmvnorm > ### Keywords: distribution > > ### ** Examples > > > n <- 5 > mean <- rep(0, 5) > lower <- rep(-1, 5) > upper <- rep(3, 5) > corr <- diag(5) > corr[lower.tri(corr)] <- 0.5 > corr[upper.tri(corr)] <- 0.5 > prob <- pmvnorm(lower, upper, mean, corr) > print(prob) [1] 0.5800051 attr(,"error") [1] 0.0002696831 attr(,"msg") [1] "Normal Completion" > > stopifnot(pmvnorm(lower=-Inf, upper=3, mean=0, sigma=1) == pnorm(3)) > > a <- pmvnorm(lower=-Inf,upper=c(.3,.5),mean=c(2,4),diag(2)) > > stopifnot(round(a,16) == round(prod(pnorm(c(.3,.5),c(2,4))),16)) > > a <- pmvnorm(lower=-Inf,upper=c(.3,.5,1),mean=c(2,4,1),diag(3)) > > stopifnot(round(a,16) == round(prod(pnorm(c(.3,.5,1),c(2,4,1))),16)) > > # Example from R News paper (original by Genz, 1992): > > m <- 3 > sigma <- diag(3) > sigma[2,1] <- 3/5 > sigma[3,1] <- 1/3 > sigma[3,2] <- 11/15 > pmvnorm(lower=rep(-Inf, m), upper=c(1,4,2), mean=rep(0, m), corr=sigma) [1] 0.8279847 attr(,"error") [1] 2.658133e-07 attr(,"msg") [1] "Normal Completion" > > # Correlation and Covariance > > a <- pmvnorm(lower=-Inf, upper=c(2,2), sigma = diag(2)*2) > b <- pmvnorm(lower=-Inf, upper=c(2,2)/sqrt(2), corr=diag(2)) > stopifnot(all.equal(round(a,5) , round(b, 5))) > > > > > base::assign(".dptime", (proc.time() - get(".ptime", pos = "CheckExEnv")), pos = "CheckExEnv") > base::cat("pmvnorm", base::get(".format_ptime", pos = 'CheckExEnv')(get(".dptime", pos = "CheckExEnv")), "\n", file=base::get(".ExTimings", pos = 'CheckExEnv'), append=TRUE, sep="\t") > cleanEx() > nameEx("pmvt") > ### * pmvt > > flush(stderr()); flush(stdout()) > > base::assign(".ptime", proc.time(), pos = "CheckExEnv") > ### Name: pmvt > ### Title: Multivariate t Distribution > ### Aliases: pmvt > ### Keywords: distribution > > ### ** Examples > > > n <- 5 > lower <- -1 > upper <- 3 > df <- 4 > corr <- diag(5) > corr[lower.tri(corr)] <- 0.5 > delta <- rep(0, 5) > prob <- pmvt(lower=lower, upper=upper, delta=delta, df=df, corr=corr) > print(prob) [1] 0.5063832 attr(,"error") [1] 0.0002426557 attr(,"msg") [1] "Normal Completion" > > pmvt(lower=-Inf, upper=3, df = 3, sigma = 1) == pt(3, 3) upper TRUE > > # Example from R News paper (original by Edwards and Berry, 1987) > > n <- c(26, 24, 20, 33, 32) > V <- diag(1/n) > df <- 130 > C <- c(1,1,1,0,0,-1,0,0,1,0,0,-1,0,0,1,0,0,0,-1,-1,0,0,-1,0,0) > C <- matrix(C, ncol=5) > ### scale matrix > cv <- C %*% V %*% t(C) > ### correlation matrix > dv <- t(1/sqrt(diag(cv))) > cr <- cv * (t(dv) %*% dv) > delta <- rep(0,5) > > myfct <- function(q, alpha) { + lower <- rep(-q, ncol(cv)) + upper <- rep(q, ncol(cv)) + pmvt(lower=lower, upper=upper, delta=delta, df=df, + corr=cr, abseps=0.0001) - alpha + } > > ### uniroot for this simple problem > round(uniroot(myfct, lower=1, upper=5, alpha=0.95)$root, 3) [1] 2.561 > > # compare pmvt and pmvnorm for large df: > > a <- pmvnorm(lower=-Inf, upper=1, mean=rep(0, 5), corr=diag(5)) > b <- pmvt(lower=-Inf, upper=1, delta=rep(0, 5), df=rep(300,5), + corr=diag(5)) Warning in if (df < 0) stop("cannot compute multivariate t distribution with ", : the condition has length > 1 and only the first element will be used Warning in if (isInf(df)) df <- 0 : the condition has length > 1 and only the first element will be used > a [1] 0.4215702 attr(,"error") [1] 0 attr(,"msg") [1] "Normal Completion" > b [1] 0.4211423 attr(,"error") [1] 2.31377e-06 attr(,"msg") [1] "Normal Completion" > > stopifnot(round(a, 2) == round(b, 2)) > > # correlation and scale matrix > > a <- pmvt(lower=-Inf, upper=2, delta=rep(0,5), df=3, + sigma = diag(5)*2) > b <- pmvt(lower=-Inf, upper=2/sqrt(2), delta=rep(0,5), + df=3, corr=diag(5)) > attributes(a) <- NULL > attributes(b) <- NULL > a [1] 0.5653944 > b [1] 0.5654 > stopifnot(all.equal(round(a,3) , round(b, 3))) > > a <- pmvt(0, 1,df=10) > attributes(a) <- NULL > b <- pt(1, df=10) - pt(0, df=10) > stopifnot(all.equal(round(a,10) , round(b, 10))) > > > > > base::assign(".dptime", (proc.time() - get(".ptime", pos = "CheckExEnv")), pos = "CheckExEnv") > base::cat("pmvt", base::get(".format_ptime", pos = 'CheckExEnv')(get(".dptime", pos = "CheckExEnv")), "\n", file=base::get(".ExTimings", pos = 'CheckExEnv'), append=TRUE, sep="\t") > cleanEx() > nameEx("qmvnorm") > ### * qmvnorm > > flush(stderr()); flush(stdout()) > > base::assign(".ptime", proc.time(), pos = "CheckExEnv") > ### Name: qmvnorm > ### Title: Quantiles of the Multivariate Normal Distribution > ### Aliases: qmvnorm > ### Keywords: distribution > > ### ** Examples > > qmvnorm(0.95, sigma = diag(2), tail = "both") $quantile [1] 2.236358 $f.quantile [1] -1.681424e-06 attr(,"message") [1] "Normal Completion" > > > > base::assign(".dptime", (proc.time() - get(".ptime", pos = "CheckExEnv")), pos = "CheckExEnv") > base::cat("qmvnorm", base::get(".format_ptime", pos = 'CheckExEnv')(get(".dptime", pos = "CheckExEnv")), "\n", file=base::get(".ExTimings", pos = 'CheckExEnv'), append=TRUE, sep="\t") > cleanEx() > nameEx("qmvt") > ### * qmvt > > flush(stderr()); flush(stdout()) > > base::assign(".ptime", proc.time(), pos = "CheckExEnv") > ### Name: qmvt > ### Title: Quantiles of the Multivariate t Distribution > ### Aliases: qmvt > ### Keywords: distribution > > ### ** Examples > > ## basic evaluation > qmvt(0.95, df = 16, tail = "both") $quantile [1] 2.119905 $f.quantile [1] 0.975 > > ## check behavior for df=0 and df=Inf > Sigma <- diag(2) > set.seed(29) > q0 <- qmvt(0.95, sigma = Sigma, df = 0, tail = "both")$quantile > set.seed(29) > q8 <- qmvt(0.95, sigma = Sigma, df = Inf, tail = "both")$quantile > set.seed(29) > qn <- qmvnorm(0.95, sigma = Sigma, tail = "both")$quantile > stopifnot(identical(q0, q8), + isTRUE(all.equal(q0, qn, tol = (.Machine$double.eps)^(1/3)))) > > ## if neither sigma nor corr are provided, corr = 1 is used internally > df <- 0 > set.seed(29) > qt95 <- qmvt(0.95, df = df, tail = "both")$quantile > set.seed(29) > qt95.c <- qmvt(0.95, df = df, corr = 1, tail = "both")$quantile > set.seed(29) > qt95.s <- qmvt(0.95, df = df, sigma = 1, tail = "both")$quantile > stopifnot(identical(qt95, qt95.c), + identical(qt95, qt95.s)) > > df <- 4 > set.seed(29) > qt95 <- qmvt(0.95, df = df, tail = "both")$quantile > set.seed(29) > qt95.c <- qmvt(0.95, df = df, corr = 1, tail = "both")$quantile > set.seed(29) > qt95.s <- qmvt(0.95, df = df, sigma = 1, tail = "both")$quantile > stopifnot(identical(qt95, qt95.c), + identical(qt95, qt95.s)) > > > > base::assign(".dptime", (proc.time() - get(".ptime", pos = "CheckExEnv")), pos = "CheckExEnv") > base::cat("qmvt", base::get(".format_ptime", pos = 'CheckExEnv')(get(".dptime", pos = "CheckExEnv")), "\n", file=base::get(".ExTimings", pos = 'CheckExEnv'), append=TRUE, sep="\t") > ### *