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Moment bounds of PH distributions with infinite or finite support based on the steepest increase property

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  22 July 2019

Qi-Ming He ,
Gábor Horváth ,
Illés Horváth  and
Miklós Telek
Qi-Ming He*
Affiliation:
University of Waterloo
Gábor Horváth*
Affiliation:
Budapest University of Technology and Economics
Illés Horváth*
Affiliation:
MTA-BME Information Systems Research Group
Miklós Telek*
Affiliation:
Budapest University of Technology and Economics
*
*Postal address: Department of Management Sciences, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email address: [email protected]
**Postal address: Department of Networked Systems and Services, Budapest University of Technology and Economics, PO Box 91, 1521 Budapest, Hungary.
****Postal address: MTA-BME Information Systems Research Group, PO Box 91, 1521 Budapest, Hungary. Email address: [email protected]
**Postal address: Department of Networked Systems and Services, Budapest University of Technology and Economics, PO Box 91, 1521 Budapest, Hungary.
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Abstract

The steepest increase property of phase-type (PH) distributions was first proposed in O’Cinneide (1999) and proved in O’Cinneide (1999) and Yao (2002), but since then has received little attention in the research community. In this work we demonstrate that the steepest increase property can be applied for proving previously unknown moment bounds of PH distributions with infinite or finite support. Of special interest are moment bounds free of specific PH representations except the size of the representation. For PH distributions with infinite support, it is shown that such a PH distribution is stochastically smaller than or equal to an Erlang distribution of the same size. For PH distributions with finite support, a class of distributions which was introduced and investigated in Ramaswami and Viswanath (2014), it is shown that the squared coefficient of variation of a PH distribution with finite support is greater than or equal to 1/(m(m + 2)), where m is the size of its PH representation.

Keywords

Phase-type distributioninfinite supportfinite supportmoment bound

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62E15: Exact distribution theory
Type
Original Article
Information
Advances in Applied Probability , Volume 51 , Issue 1 , March 2019 , pp. 168 - 183
Copyright
© Applied Probability Trust 2019 

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References

Aldous, D. and Shepp, L. (1987). The least variable phase type distribution is Erlang. Commun. Statist. Stoch. Models 3, 467–473.Google Scholar
He, Q.-M., Zhang, H., and Vera, J. (2013). Majorization and extremal PH-distributions. In Matrix-Analytic Methods in Stochastic Models (Springer Proc. Math. Statist. 27), eds Latouche, G. et al., Springer, New York, pp. 107–121.CrossRefGoogle Scholar
Neuts, M. F. (1975). Probability distributions of phase type. In Liber Amicorum Prof. Emeritus H. Florin, University of Louvain, Belgium, pp. 173–206.Google Scholar
Neuts, M. F. (1981). Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. Johns Hopkins University Press, Baltimore.Google Scholar
O’cinneide, C. A. (1989). On non-uniqueness of representations of phase-type distributions. Commun. Statist. Stoch. Models 5, 247–259.CrossRefGoogle Scholar
O’CINNEIDE, C. A. (1990). Characterization of phase-type distributions. Commun. Statist. Stoch. Models 6, 1–57.Google Scholar
O’cinneide, C. A. (1991). Phase-type distributions and majorization. Ann. Appl. Prob. 1, 219–227.Google Scholar
O’cinneide, C. A. (1999). Phase-type distributions: open problems and a few properties. Commun. Statist. Stoch. Models, 15, 731–757,Google Scholar
Ramaswami, V. and Viswanath, N. C. (2014). Phase type distributions with finite support. Stoch. Models, 30, 576–597.Google Scholar
Stoyan, D. (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, Chichester.Google Scholar
Telek, M. (2000). The minimal coefficient of variation of discrete phase type distributions. Proc. 3rd Internat. Conf. on Matrix Analytic Methods in Stochastic Models, eds Latouche, G. and G. Taylor, P., Notable Publications, New Jersey.Google Scholar
Yao, R. (2002). A proof of the steepest increase conjecture of a phase-type density. Stoch. Models, 18, 1–6.Google Scholar