Mvt {mvtnorm}R Documentation

The Multivariate t Distribution

Description

These functions provide information about the multivariate t distribution with non-centrality parameter (or mode) delta, covariance matrix sigma and degrees of freedom df. dmvt gives the density and rmvt generates random deviates.

Usage

rmvt(n, sigma = diag(2), df = 1, delta = rep(0, nrow(sigma)),
     type = c("shifted", "Kshirsagar"))
dmvt(x, delta, sigma, df = 1, log = TRUE,
     type = "shifted")

Arguments

x Vector or matrix of quantiles. If x is a matrix, each row is taken to be a quantile.
n Number of observations.
delta the vector of noncentrality parameters of length n, for type = "shifted" delta specifies the mode.
sigma Covariance matrix, default is diag(ncol(x)).
df degree of freedom as integer.
log Logical; if TRUE, densities d are given as log(d).
type type of the noncentral multivariate t distribution to be computed. 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. 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. Note that the defaults differ from the default in pmvt (for reasons of backward compatibility).

Details

For type = "shifted" the following density is implemented

c(1+(x-δ)'S^{-1}(x-δ)/ν)^{-(ν+m)/2},

where

c = Γ((ν+m)/2)/((π ν)^{m/2}Γ(ν/2)|S|^{1/2}),

here S is a positive definite symmetric matrix (which might be the correlation or the covariance matrix), delta is the non-centrality vector and ν are the degrees of freedom.

See Also

pmvt and qmvt

Examples


  dmvt(x=c(0,0), sigma = diag(2))
  x <- rmvt(n=100, sigma = diag(2), df = 3)
  plot(x)


[Package mvtnorm version 0.9-96 Index]