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

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

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 \(\textbf{S}\).

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 \(k \times k\) positive-definite scale
matrix \(\textbf{S}\), 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.

`dmvt`

gives the density and
`rmvt`

generates random deviates.

Application: Continuous Multivariate

Density: $$p(\theta) = \frac{\Gamma[(\nu+k)/2]}{\Gamma(\nu/2)\nu^{k/2}\pi^{k/2}|\Sigma|^{1/2}[1 + (1/\nu)(\theta-\mu)^{\mathrm{T}} \Sigma^{-1} (\theta-\mu)]^{(\nu+k)/2}}$$

Inventor: Unknown (to me, anyway)

Notation 1: \(\theta \sim \mathrm{t}_k(\mu, \Sigma, \nu)\)

Notation 2: \(p(\theta) = \mathrm{t}_k(\theta | \mu, \Sigma, \nu)\)

Parameter 1: location vector \(\mu\)

Parameter 2: positive-definite \(k \times k\) scale matrix \(\Sigma\)

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

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

Variance: \(var(\theta) = \frac{\nu}{\nu - 2} \Sigma\), 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 \(k \times k\) scale matrix \(\textbf{S}\), which must be positive-definite. When degrees of freedom \(\nu=1\), this is the multivariate Cauchy distribution.

```
# NOT RUN {
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)
# }
```

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