# The Multivariate t Distribution These functions provide information about the multivariate $t$ distribution with non-centrality parameter (or mode) `delta`, scale matrix `sigma` and degrees of freedom `df`. `dmvt` gives the density and `rmvt` generates random deviates. ```r rmvt(n, sigma = diag(2), df = 1, delta = rep(0, nrow(sigma)), type = c("shifted", "Kshirsagar"), ...) dmvt(x, delta = rep(0, p), sigma = diag(p), df = 1, log = TRUE, type = "shifted", checkSymmetry = TRUE) ``` ## Arguments - `x`: vector or matrix of quantiles. If `x` is a matrix, each row is taken to be a quantile. - `n`: number of observations. - `delta`: the vector of noncentrality parameters of length n, for `type = "shifted"` delta specifies the mode. - `sigma`: scale matrix, defaults to `diag(ncol(x))`. - `df`: degrees of freedom. `df = 0` or `df = Inf` corresponds to the multivariate normal distribution. - `log`: `logical` indicating whether densities $d$ are given as $log(d)$. - `type`: type of the noncentral multivariate $t$ distribution. `type = "Kshirsagar"` corresponds to formula (1.4) in Genz and Bretz (2009) (see also Chapter 5.1 in Kotz and Nadarajah (2004)). This is the noncentral t-distribution needed for calculating the power of multiple contrast tests under a normality assumption. `type = "shifted"` corresponds to the formula right before formula (1.4) in Genz and Bretz (2009) (see also formula (1.1) in Kotz and Nadarajah (2004)). It is a location shifted version of the central t-distribution. This noncentral multivariate $t$ distribution appears for example as the Bayesian posterior distribution for the regression coefficients in a linear regression. In the central case both types coincide. Note that the defaults differ from the default in `pmvt()` (for reasons of backward compatibility). - `checkSymmetry`: logical; if `FALSE`, skip checking whether the covariance matrix is symmetric or not. This will speed up the computation but may cause unexpected outputs when ill-behaved `sigma` is provided. The default value is `TRUE`. - ``...``: additional arguments to `rmvnorm()`, for example `method`. ## Details If $X$ denotes a random vector following a $t$ distribution with location vector $0$ and scale matrix $Sigma$ (written c("$X ~ t_nu(0,\n$", "$ Sigma)$")), the scale matrix (the argument `sigma`) is not equal to the covariance matrix $Cov(X)$ of $X$. If the degrees of freedom $nu$ (the argument `df`) is larger than 2, then c("$Cov(X)=Sigma\n$", "$ nu/(nu-2)$"). Furthermore, in this case the correlation matrix $Cor(X)$ equals the correlation matrix corresponding to the scale matrix $Sigma$ (which can be computed with `cov2cor()`). Note that the scale matrix is sometimes referred to as dispersion matrix ; see McNeil, Frey, Embrechts (2005, p. 74). For `type = "shifted"` the density $$ c(1+(x-\delta)'S^{-1}(x-\delta)/\nu)^{-(\nu+m)/2} $$ is implemented, where $$ c = \Gamma((\nu+m)/2)/((\pi \nu)^{m/2}\Gamma(\nu/2)|S|^{1/2}), $$ $S$ is a positive definite symmetric matrix (the matrix `sigma` above), $delta$ is the non-centrality vector and $nu$ are the degrees of freedom. `df=0` historically leads to the multivariate normal distribution. From a mathematical point of view, rather `df=Inf` corresponds to the multivariate normal distribution. This is (now) also allowed for `rmvt()` and `dmvt()`. Note that `dmvt()` has default `log = TRUE`, whereas `dmvnorm()` has default `log = FALSE`. ## References McNeil, A. J., Frey, R., and Embrechts, P. (2005). **Quantitative Risk Management: Concepts, Techniques, Tools**. Princeton University Press. ## See Also `pmvt()` and `qmvt()` ## Examples ```r ## basic evaluation dmvt(x = c(0,0), sigma = diag(2)) ## check behavior for df=0 and df=Inf x <- c(1.23, 4.56) mu <- 1:2 Sigma <- diag(2) x0 <- dmvt(x, delta = mu, sigma = Sigma, df = 0) # default log = TRUE! x8 <- dmvt(x, delta = mu, sigma = Sigma, df = Inf) # default log = TRUE! xn <- dmvnorm(x, mean = mu, sigma = Sigma, log = TRUE) stopifnot(identical(x0, x8), identical(x0, xn)) ## X ~ t_3(0, diag(2)) x <- rmvt(100, sigma = diag(2), df = 3) # t_3(0, diag(2)) sample plot(x) ## X ~ t_3(mu, Sigma) n <- 1000 mu <- 1:2 Sigma <- matrix(c(4, 2, 2, 3), ncol=2) set.seed(271) x <- rep(mu, each=n) + rmvt(n, sigma=Sigma, df=3) plot(x) ## Note that the call rmvt(n, mean=mu, sigma=Sigma, df=3) does *not* ## give a valid sample from t_3(mu, Sigma)! [and thus throws an error] try(rmvt(n, mean=mu, sigma=Sigma, df=3)) ## df=Inf correctly samples from a multivariate normal distribution set.seed(271) x <- rep(mu, each=n) + rmvt(n, sigma=Sigma, df=Inf) set.seed(271) x. <- rmvnorm(n, mean=mu, sigma=Sigma) stopifnot(identical(x, x.)) ```

Mvt function