dtmvt: Truncated Multivariate Student t Density
In tmvtnorm: Truncated Multivariate Normal and Student t Distribution

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/dtmvt.R

This function provides the joint density function for the truncated multivariate Student t distribution with mean vector equal to mean, covariance matrix sigma, degrees of freedom parameter df and lower and upper truncation points lower and upper.

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dtmvt(x, mean = rep(0, nrow(sigma)), sigma = diag(length(mean)), df = 1, 
lower = rep(-Inf, length = length(mean)), 
upper = rep(Inf, length = length(mean)), log = FALSE)

`x`	Vector or matrix of quantiles. If `x` is a matrix, each row is taken to be a quantile.
`mean`	Mean vector, default is `rep(0, nrow(sigma))`.
`sigma`	Covariance matrix, default is `diag(length(mean))`.
`df`	degrees of freedom parameter
`lower`	Vector of lower truncation points, default is `rep(-Inf, length = length(mean))`.
`upper`	Vector of upper truncation points, default is `rep( Inf, length = length(mean))`.
`log`	Logical; if `TRUE`, densities d are given as log(d).

The Truncated Multivariate Student t Distribution is a conditional Multivariate Student t distribution subject to (linear) constraints a ≤ \bold{x} ≤ b.

The density of the p-variate Multivariate Student t distribution with nu degrees of freedom is

f(\bold{x}) = \frac{Γ((ν + p)/2)}{(πν)^{p/2} Γ(ν/2) \|Σ\|^{1/2}} [ 1 + \frac{1}{ν} (x - μ)^T Σ^{-1} (x - μ) ]^{- (ν + p) / 2}

The density of the truncated distribution f_{a,b}(x) with constraints a <= x <= b is accordingly

f_{a,b}(x) = \frac{f(\bold{x})} {P(a ≤ x ≤ b)}

a numeric vector with density values

Stefan Wilhelm wilhelm@financial.com

Geweke, J. F. (1991) Efficient simulation from the multivariate normal and Student-t distributions subject to linear constraints and the evaluation of constraint probabilities. http://www.biz.uiowa.edu/faculty/jgeweke/papers/paper47/paper47.pdf

Samuel Kotz, Saralees Nadarajah (2004). Multivariate t Distributions and Their Applications. Cambridge University Press

ptmvt and rtmvt for probabilities and random number generation in the truncated case, see dmvt, rmvt and pmvt for the untruncated multi-t distribution.

# Example

x1 <- seq(-2, 3, by=0.1)
x2 <- seq(-2, 3, by=0.1)

mean <- c(0,0)
sigma <- matrix(c(1, -0.5, -0.5, 1), 2, 2)
lower <- c(-1,-1)


density <- function(x)
{
	z=dtmvt(x, mean=mean, sigma=sigma, lower=lower)
	z
}

fgrid <- function(x, y, f)
{
	z <- matrix(nrow=length(x), ncol=length(y))
	for(m in 1:length(x)){
		for(n in 1:length(y)){
			z[m,n] <- f(c(x[m], y[n]))
		}
	}
	z
}

# compute multivariate-t density d for grid
d <- fgrid(x1, x2, function(x) dtmvt(x, mean=mean, sigma=sigma, lower=lower))

# compute multivariate normal density d for grid
d2 <- fgrid(x1, x2, function(x) dtmvnorm(x, mean=mean, sigma=sigma, lower=lower))

# plot density as contourplot
contour(x1, x2, d, nlevels=5, main="Truncated Multivariate t Density", 
		xlab=expression(x[1]), ylab=expression(x[2]))

contour(x1, x2, d2, nlevels=5, add=TRUE, col="red")
abline(v=-1, lty=3, lwd=2)
abline(h=-1, lty=3, lwd=2)