Equivalence and Separation for Multivariate Matrix-Exponential and Phase-Type Distribution Classes

TL;DR

This paper resolves open questions about the equivalence and separation of multivariate matrix-exponential and phase-type distribution classes, providing new representations and examples.

Contribution

It proves the equivalence of MVME and MME* classes and demonstrates strict separation between MPH* and MVPH in multivariate cases, with new factorization conditions.

Findings

MVME coincides with MME* class for matrix-exponential distributions.

MPH* is strictly contained in MVPH from trivariate case onward.

A Wishart trace distribution belongs to MVPH but not to MPH*.

Abstract

We resolve two questions left open by Bladt and Nielsen (2010) concerning multivariate families of matrix-exponential and phase-type distributions. First, in the matrix-exponential case, the projection-defined class MVME coincides with Kulkarni's algebraic class MME*. Our proof combines a multivariate state-space realization theorem with elementary augmentations that put the realization into Kulkarni's form. Thus every proper rational multivariate Laplace transform has a finite-dimensional Kulkarni-type representation once Markovian sign constraints are removed. Second, in the phase-type setting, the inclusion of MPH* in MVPH is strict from the trivariate case onward. The separation is obtained through a factorization condition for MPH* that appears not to have been previously identified in the PH literature. A Wishart trace distribution belongs to MVPH but fails this condition, hence providing the required example outside MPH*. The example also shows that projection-based multivariate phase-type laws may have density and support geometry that are absent from the usual univariate theory.