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On matrix exponential distributions

Part of: Foundations of probability theory Basic linear algebra

Published online by Cambridge University Press:  01 July 2016

Qi-Ming He  and
Hanqin Zhang
Qi-Ming He*
Affiliation:
Dalhousie University
Hanqin Zhang*
Affiliation:
Chinese Academy of Sciences
*
∗ Postal address: Department of Industrial Engineering, Dalhousie University, Halifax, NS B3J 2X4, Canada. Email address: qi-ming.he@dal.ca
∗∗ Postal address: Institute of Applied Mathematics, Chinese Academy of Sciences, 100080 Beijing, P. R. China. Email address: hanqin@amt.ac.cn
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Abstract

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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.

Keywords

Hankel matrixmatrix exponential distributionphase-type distributionmatrix-analytic methods

MSC classification

Primary: 60A99: None of the above, but in this section
Secondary: 15A18: Eigenvalues, singular values, and eigenvectors
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 271 - 292
Copyright
Copyright © Applied Probability Trust 2007 

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