# Multivariate Student t distribution Density and random generation for the multivariate Student t distribution with location equal to `mu`, precision matrix equal to `Q` (or scale matrix equal to `Sigma`). Mentioned functions implement the multivariate Student t distribution with a density given by [REMOVE_ME]$$ %p(\boldsymbol{z}) =\frac{\Gamma\bigl(\frac{\nu+p}{2}\bigr)}{\Gamma\bigl(\frac{\nu}{2}\bigr)\,\nu^{\frac{p}{2}}\,\pi^{\frac{p}{2}}}\,\bigl|\Sigma\bigr|^{-\frac{1}{2}}\,\Bigl\{1 + \frac{(\boldsymbol{z} -\boldsymbol{\mu})'\Sigma^{-1}(\boldsymbol{z} -\boldsymbol{\mu})}{\nu}\Bigr\}^{-\frac{\nu+p}{2}},%p(z) = (Gamma((nu+p)/2)/(Gamma(nu/2) * nu^(p/2) * pi^(p/2))) *|Sigma|^(-1/2) * (1 + ((z - mu)'*Sigma^(-1)*(z - mu)) / nu)^(-(nu+p)/2), [REMOVE_ME_2]$$ where $p$ is the dimension, $nu > 0$ degrees of freedom, $mu$ the location parameter and $Sigma$ the scale matrix. For $nu > 1$, the mean in equal to $mu$, for $nu > 2$, the covariance matrix is equal to $(nu / (nu - 2)) * Sigma$. ## Description Density and random generation for the multivariate Student t distribution with location equal to `mu`, precision matrix equal to `Q` (or scale matrix equal to `Sigma`). Mentioned functions implement the multivariate Student t distribution with a density given by $$ %p(\boldsymbol{z}) =\frac{\Gamma\bigl(\frac{\nu+p}{2}\bigr)}{\Gamma\bigl(\frac{\nu}{2}\bigr)\,\nu^{\frac{p}{2}}\,\pi^{\frac{p}{2}}}\,\bigl|\Sigma\bigr|^{-\frac{1}{2}}\,\Bigl\{1 + \frac{(\boldsymbol{z} -\boldsymbol{\mu})'\Sigma^{-1}(\boldsymbol{z} -\boldsymbol{\mu})}{\nu}\Bigr\}^{-\frac{\nu+p}{2}},%p(z) = (Gamma((nu+p)/2)/(Gamma(nu/2) * nu^(p/2) * pi^(p/2))) *|Sigma|^(-1/2) * (1 + ((z - mu)'*Sigma^(-1)*(z - mu)) / nu)^(-(nu+p)/2), $$ where $p$ is the dimension, $nu > 0$ degrees of freedom, $mu$ the location parameter and $Sigma$ the scale matrix. For $nu > 1$, the mean in equal to $mu$, for $nu > 2$, the covariance matrix is equal to $(nu / (nu - 2)) * Sigma$. ```r dMVT(x, df, mu=0, Q=1, Sigma, log=FALSE) rMVT(n, df, mu=0, Q=1, Sigma) ``` ## Arguments - `df`: degrees of freedom of the multivariate Student t distribution. - `mu`: vector of the location parameter. - `Q`: precision (inverted scale) matrix of the multivariate Student t distribution. Ignored if `Sigma` is given. - `Sigma`: scale matrix of the multivariate Student t distribution. If `Sigma` is supplied, precision is computed from $Sigma$ as $Q = Sigma^{-1}$. - `n`: number of observations to be sampled. - `x`: vector or matrix of the points where the density should be evaluated. - `log`: logical; if `TRUE`, log-density is computed ## Returns Some objects. ## Value for dMVT A vector with evaluated values of the (log-)density ## Value for rMVT A list with the components: - **x**: vector or matrix with sampled values - **log.dens**: vector with the values of the log-density evaluated in the sampled values ## See Also `dt`, `Mvt`. ## Author(s) Arnošt Komárek arnost.komarek@mff.cuni.cz ## Examples ```r set.seed(1977) ### Univariate central t distribution z <- rMVT(10, df=1, mu=0, Q=1) ldz <- dMVT(z$x, df=1, log=TRUE) boxplot(as.numeric(z$x)) cbind(z$log.dens, ldz, dt(as.numeric(z$x), df=1, log=TRUE)) ### Multivariate t distribution mu <- c(1, 2, 3) Sigma <- matrix(c(1, 1, -1.5, 1, 4, 1.8, -1.5, 1.8, 9), nrow=3) Q <- chol2inv(chol(Sigma)) nu <- 3 z <- rMVT(1000, df=nu, mu=mu, Sigma=Sigma) apply(z$x, 2, mean) ## should be close to mu ((nu - 2) / nu) * var(z$x) ## should be close to Sigma dz <- dMVT(z$x, df=nu, mu=mu, Sigma=Sigma) ldz <- dMVT(z$x, df=nu, mu=mu, Sigma=Sigma, log=TRUE) ### Compare with mvtnorm package #require(mvtnorm) #ldz2 <- dmvt(z$x, sigma=Sigma, df=nu, delta=mu, log=TRUE) #plot(z$log.dens, ldz2, pch=21, col="red3", bg="orange", xlab="mixAK", ylab="mvtnorm") #plot(ldz, ldz2, pch=21, col="red3", bg="orange", xlab="mixAK", ylab="mvtnorm") ```

MVT function