# Truncated Multivariate Student t Density 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`. ```r 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) ``` ## Arguments - `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). ## Details The Truncated Multivariate Student t Distribution is a conditional Multivariate Student t distribution subject to (linear) constraints $a \le \bold{x} \le b$. The density of the $p$-variate Multivariate Student t distribution with $nu$ degrees of freedom is $$ f(\bold{x}) = \frac{\Gamma((\nu + p)/2)}{(\pi\nu)^{p/2} \Gamma(\nu/2) \|\Sigma\|^{1/2}}[ 1 + \frac{1}{\nu} (x - \mu)^T \Sigma^{-1} (x - \mu) ]^{- (\nu + p) / 2} $$ The density of the truncated distribution $f_{a,b}(x)$ with constraints $a <= x <= b$ is accordingly $$ f_{a,b}(x) = \frac{f(\bold{x})} {P(a \le x \le b)} $$ ## Returns a numeric vector with density values ## See Also `ptmvt` and `rtmvt` for probabilities and random number generation in the truncated case, see `dmvt`, `rmvt` and `pmvt` for the untruncated multi-t distribution. ## References 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](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** ## Author(s) Stefan Wilhelm wilhelm@financial.com ## Examples ```r # 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) ```

dtmvt function