Multivariate t distribution — mvt • RTMBdist

Density and and random generation for the multivariate t distribution

Usage

dmvt(x, mu, Sigma, df, log = FALSE)

rmvt(n, mu, Sigma, df)

Arguments

x: vector or matrix of quantiles
mu: vector or matrix of location parameters (mean if df > 1)
Sigma: positive definite scale matrix (proportional to the covariance matrix if df > 2)
df: degrees of freedom; must be positive
log: logical; if TRUE, densities $p$ are returned as $\log(p)$.
n: number of random values to return.

Value

dmvt gives the density, rmvt generates random deviates.

Details

This implementation of dmvt allows for automatic differentiation with RTMB.

Note: for df $\le 1$ the mean is undefined, and for df $\le 2$ the covariance is infinite. For df > 2, the covariance is df/(df-2) * Sigma.

$$f(\mathbf{x};\,\boldsymbol{\mu},\Sigma,\nu) = \frac{\Gamma((\nu+d)/2)}{\Gamma(\nu/2)\,(\nu\pi)^{d/2}\,|\Sigma|^{1/2}} \left(1 + \frac{(\mathbf{x}-\boldsymbol{\mu})^\top \Sigma^{-1}(\mathbf{x}-\boldsymbol{\mu})}{\nu}\right)^{-(\nu+d)/2},$$ where $d$ is the dimension.

Examples

# single mu
mu <- c(1,2,3)
Sigma <- diag(c(1,1,1))
df <- 5
x <- rmvt(2, mu, Sigma, df)
d <- dmvt(x, mu, Sigma, df)
# vectorised over mu
mu <- rbind(c(1,2,3), c(0, 0.5, 1))
x <- rmvt(2, mu, Sigma, df)
d <- dmvt(x, mu, Sigma, df)