# dist.Multivariate.t: Multivariate t Distribution In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These functions provide the density and random number generation for the multivariate t distribution, otherwise called the multivariate Student distribution.

## Usage

 ```1 2``` ```dmvt(x, mu, S, df=Inf, log=FALSE) rmvt(n=1, mu, S, df=Inf) ```

## Arguments

 `x` This is either a vector of length k or a matrix with a number of columns, k, equal to the number of columns in scale matrix S. `n` This is the number of random draws. `mu` This is a numeric vector or matrix representing the location parameter,mu (the mean vector), of the multivariate distribution (equal to the expected value when `df > 1`, otherwise represented as nu > 1). When a vector, it must be of length k, or must have k columns as a matrix, as defined above. `S` This is a k x k positive-definite scale matrix S, such that `S*df/(df-2)` is the variance-covariance matrix when `df > 2`. A vector of length 1 is also allowed (in this case, k=1 is set). `df` This is the degrees of freedom, and is often represented with nu. `log` Logical. If `log=TRUE`, then the logarithm of the density is returned.

## Details

• Application: Continuous Multivariate

• Density:

p(theta) = Gamma[(nu+k)/2] / {Gamma(nu/2)nu^(k/2)pi^(k/2)|Sigma|^(1/2)[1 + (1/nu)(theta-mu)^T*Sigma^(-1)(theta-mu)]^[(nu+k)/2]}

• Inventor: Unknown (to me, anyway)

• Notation 1: theta ~ t[k](mu, Sigma, nu)

• Notation 2: p(theta) = t[k](theta | mu, Sigma, nu)

• Parameter 1: location vector mu

• Parameter 2: positive-definite k x k scale matrix Sigma

• Parameter 3: degrees of freedom nu > 0 (df in the functions)

• Mean: E(theta) = mu, for nu > 1, otherwise undefined

• Variance: var(theta) = (nu / (nu - 2))*Sigma, for nu > 2

• Mode: mode(theta) = mu

The multivariate t distribution, also called the multivariate Student or multivariate Student t distribution, is a multidimensional extension of the one-dimensional or univariate Student t distribution. A random vector is considered to be multivariate t-distributed if every linear combination of its components has a univariate Student t-distribution. This distribution has a mean parameter vector mu of length k, and a k x k scale matrix S, which must be positive-definite. When degrees of freedom nu=1, this is the multivariate Cauchy distribution.

## Value

`dmvt` gives the density and `rmvt` generates random deviates.

## Author(s)

Statisticat, LLC. [email protected]

`dinvwishart`, `dmvc`, `dmvcp`, `dmvtp`, `dst`, `dstp`, and `dt`.
 ``` 1 2 3 4 5 6 7 8 9 10``` ```library(LaplacesDemon) x <- seq(-2,4,length=21) y <- 2*x+10 z <- x+cos(y) mu <- c(1,12,2) S <- matrix(c(1,2,0,2,5,0.5,0,0.5,3), 3, 3) df <- 4 f <- dmvt(cbind(x,y,z), mu, S, df) X <- rmvt(1000, c(0,1,2), S, 5) joint.density.plot(X[,1], X[,2], color=TRUE) ```