Student-t (univariate)

Parameters

Mean (\(\mu \)), Variance of underlying Normal (\(\sigma ^{2}\)) or of the distribution itself (\(s^{2}\)), Degrees of freedom (\(\nu \))

Kernel

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

or

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

Support

\(\left( -\infty ,\infty \right) \)

Mean

\(\mu \)

Variance

\(\sigma ^{2}\nu /\left( \nu -2\right) \) or \(s^{2}\)

Main Uses

Small sample distributions of univariate statistics. Fatter-tailed alternative to Normal for error processes. Prior, exact and approximate posteriors for parameters with unlimited ranges.

Density Function

%TDENSITY(x,nu) is the (non-logged) density function for a standard (\(\mu =0,\sigma ^{2}=1\)) t.

%LOGTDENSITY(ssquared,u,nu) is the log density based upon the \(s^{2}\) parameterization.
 

%LOGTDENSITYSTD(sigmasq,x-mu,nu) is the log density based upon the \(\sigma ^{2}\) parameterization.

Use %LOGTDENSITY(ssquared,x-mu,nu) to compute \(\log f\left( x|\mu ,s^{2},\nu\right) \) and

%LOGTDENSITYSTD(sigmasq,x-mu,nu) to compute \(\log f\left( x|\mu ,\sigma ^{2},\nu \right) \).

CDF

%TCDF(x,nu) is the CDF for a standard t.

%TTEST(x,nu) is the two-tailed probability of a standard t

Inverse CDF

%INVTCDF(p,nu) is the inverse CDF for a standard t.

%INVTTEST(p,nu) is the critical value for a two-tailed standard t test.

Random Draws

%RANT(nu) draws one or more (depending upon the target) standard t's with independent numerators and a common denominator. To get a draw from a t density with variance ssquared and nu degrees of freedom, use %RANT(nu)*sqrt(ssquared*(nu-2.)/nu).