Advances in Applied Probability

On matrix exponential distributions

Qi-Ming He and Hanqin Zhang

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Abstract

In this paper we introduce certain Hankel matrices that can be used to study ME (matrix exponential) distributions, in particular to compute their ME orders. The Hankel matrices for a given ME probability distribution can be constructed if one of the following five types of information about the distribution is available: (i) an ME representation, (ii) its moments, (iii) the derivatives of its distribution function, (iv) its Laplace-Stieltjes transform, or (v) its distribution function. Using the Hankel matrices, a necessary and sufficient condition for a probability distribution to be an ME distribution is found and a method of computing the ME order of the ME distribution developed. Implications for the PH (phase-type) order of PH distributions are examined. The relationship between the ME order, the PH order, and a lower bound on the PH order given by Aldous and Shepp (1987) is discussed in numerical examples.

Article information

Source
Adv. in Appl. Probab. Volume 39, Number 1 (2007), 271-292.

Dates
First available: 30 March 2007

Permanent link to this document
http://projecteuclid.org/euclid.aap/1175266478

Digital Object Identifier
doi:10.1239/aap/1175266478

Mathematical Reviews number (MathSciNet)
MR2307880

Zentralblatt MATH identifier
1114.60013

Subjects
Primary: 60A99: None of the above, but in this section
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors

Keywords
Hankel matrix matrix exponential distribution phase-type distribution matrix-analytic methods

Citation

He, Qi-Ming; Zhang, Hanqin. On matrix exponential distributions. Advances in Applied Probability 39 (2007), no. 1, 271--292. doi:10.1239/aap/1175266478. http://projecteuclid.org/euclid.aap/1175266478.


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