Student-t (multivariate)

Parameters

Mean (\(\mu \)), covariance matrix of the underlying multivariate Normal (\(\Sigma \)) or of the distribution itself (\(\mathbf{S}\)), Degrees of freedom (\(\nu \))

Kernel

\(\left( 1+\left( x-\mu \right) ^{\prime }\Sigma ^{-1}\left(x-\mu\right) /\nu \right) ^{-\left( \nu +n\right) /2}\)

or

\(\left( 1+\left( x-\mu \right) ^{\prime }\mathbf{S}^{-1}\left( x-\mu \right)/\left( \nu -2\right) \right) ^{-\left( \nu +n\right) /2}\)

Support

\(\mathbb{R}^{n}\)

Mean

\(\mu \)

Covariance Matrix

\(\Sigma \dfrac{\nu }{\left( \nu -2\right) }\) or \(\mathbf{S}\)

Main Uses

Prior, exact and approximate posteriors for sets of parameters with unlimited ranges.

Density Function

%LOGTDENSITY(s,u,nu) is the log density based upon the \(\mathbf{S}\) parameterization. Use %LOGTDENSITY(s,x-mu,nu) to compute \(\log f\left( x|\mu ,\mathbf{S},\nu \right) \)

%LOGTDENSITYSTD(sigma,u,nu) is the log density based upon the \(\mathbf{\Sigma}\) parameterization. Use %LOGTDENSITYSTD(sigma,u,nu) to compute \(\log f\left( x|\mu ,\Sigma ,\nu \right) \).

The same functions work for both univariate and multivariate distributions.

Random Draws

%RANT(nu) or %RANMVT(fsigma,nu)

%RANT(nu) draws one or more (depending upon the target) standard t's with independent numerators and a common denominator.
 

%RANMVT(fsigma,nu) draws a multivariate t with fsigma as a "square root" of the \(\Sigma\) matrix. You would typically compute fsigma as %DECOMP(sigma). The function is parameterized this way in case you need to do many draws with a single sigma, as you can do the decomposition in advance.

To draw a multivariate t with covariance matrix S and nu degrees of freedom, use %RANMVT(%DECOMP(s*((nu-2.)/nu))).