mStudent: Multivariate t distribution

In MVT: Estimation and Testing for the Multivariate t-Distribution

Multivariate t distribution

Description

These functions provide the density and random number generation from the multivariate Student-t distribution.

Usage

dmt(x, mean = rep(0, nrow(Sigma)), Sigma = diag(length(mean)), eta = 0.25, log = FALSE)
rmt(n = 1, mean = rep(0, nrow(Sigma)), Sigma = diag(length(mean)), eta = 0.25)

Arguments

x	vector or matrix of data.
n	the number of samples requested.
mean	a vector giving the means of each variable
Sigma	a positive-definite covariance matrix
eta	shape parameter (must be in [0,1/2)). Default value is 0.25
log	logical; if TRUE, the logarithm of the density function is returned.

Details

A random vector \bold{X} = (X_1,\dots,X_p)^T has a multivariate t distribution, with a \bold{\mu} mean vector, covariance matrix \bold{\Sigma}, and 0 \leq \eta < 1/2 shape parameter, if its density function is given by

f(\bold{x}) = K_p(\eta)|\bold{\Sigma}|^{-1/2}\left\{1 + c(\eta)(\bold{x} - \bold{\mu})^T \bold{\Sigma}^{-1} (\bold{x} - \bold{\mu})\right\}^{-\frac{1}{2\eta}(1 + \eta p)}.

where

K_p(\eta) = \left(\frac{c(\eta)}{\pi}\right)^{p/2}\frac{\Gamma(\frac{1}{2\eta}(1 + \eta p))} {\Gamma(\frac{1}{2\eta})},

with c(\eta)=\eta/(1 - 2\eta). This parameterization of the multivariate t distribution is introduced mainly because \bold{\mu} and \bold{\Sigma} correspond to the mean vector and covariance matrix, respectively.

The function rmt is an interface to C routines, which make calls to subroutines from LAPACK. The matrix decomposition is internally done using the Cholesky decomposition. If Sigma is not non-negative definite then there will be a warning message.

This parameterization of the multivariate-t includes the normal distribution as a particular case when eta = 0.

Value

If x is a matrix with n rows, then dmt returns a n\times 1 vector considering each row of x as a copy from the multivariate t distribution.

If n = 1, then rmt returns a vector of the same length as mean, otherwise a matrix of n rows of random vectors.

References

Fang, K.T., Kotz, S., Ng, K.W. (1990). Symmetric Multivariate and Related Distributions. Chapman & Hall, London.

Gomez, E., Gomez-Villegas, M.A., Marin, J.M. (1998). A multivariate generalization of the power exponential family of distributions. Communications in Statistics - Theory and Methods 27, 589-600.

Examples

# covariance matrix
Sigma <- matrix(c(10,3,3,2), ncol = 2)
Sigma

# generate the sample
y <- rmt(n = 1000, Sigma = Sigma)

# scatterplot of a random bivariate t sample with mean vector
# zero and covariance matrix 'Sigma'
par(pty = "s")
plot(y, xlab = "", ylab = "")
title("bivariate t sample (eta = 0.25)", font.main = 1)

MVT documentation built on Sept. 24, 2024, 5:08 p.m.