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, scale 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

Correlation 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).

...

additional arguments to rmvnorm, for example method.

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 scale 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 0.9-9994 Index]