dtmvt | R Documentation |
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
.
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 |
mean |
Mean vector, default is |
sigma |
Covariance matrix, default is |
df |
degrees of freedom parameter |
lower |
Vector of lower truncation points,
default is |
upper |
Vector of upper truncation points,
default is |
log |
Logical; if |
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. https://www.researchgate.net/publication/2335219_Efficient_Simulation_from_the_Multivariate_Normal_and_Student-t_Distributions_Subject_to_Linear_Constraints_and_the_Evaluation_of_Constraint_Probabilities
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)
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