Closure of multivariate t and related distributions

Mathematical Properties of the Multivariate t Distribution

“…In the case p ¼ 1, (29) coincides with the entropies for the univariate Student's t and Cauchy distributions given, for example, in Lazo and Rathie [36]. Zografos [55] provided a maximum entropy characterization of (1).…”

“…Jensen [29] also studied the concentration properties of (1) via peakedness by varying its parameters. If X is multivariate normal, then the transformation X Y T(X) diminishes the peakedness.…”

“…Jensen [29] established that the class of multivariate t distributions is closed under the transform T(Á ). Specifically, assume A T A = I k , B T B = I # , and X is distributed according to (1) with zero means, correlation matrix I n , and degrees of freedom m. Under these assumptions, Jensen showed that T(X) is also distributed according to (1) with zero means, correlation matrix I k , and degrees of freedom #.…”

Estimation Methods for the Multivariate t Distribution

“…Among others, Zellner [60] and Sutradhar and Ali [54] considered (14) in the context of stock market problems. By successive integration, one can show that the marginal distribution of X j in the multivariate t model (14) is p-variate t .…”

“…The degrees of freedom parameter ν is also referred to as the shape parameter, as the peakedness of (1) may be diminished, preserved or increased by varying ν [14]. The distribution is said to be central if μ = 0.…”