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Stochastic comparison of parallel systems with Pareto components

Part of: Nonparametric inference Sampling theory, sample surveys Survival analysis and censored data Distribution theory - Probability

Published online by Cambridge University Press:  20 May 2021

Sameen Naqvi Open the ORCID record for Sameen Naqvi [Opens in a new window] ,
Weiyong Ding  and
Peng Zhao Open the ORCID record for Peng Zhao [Opens in a new window]
Sameen Naqvi
Affiliation:
Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi 502285, India. E-mail: sameen@math.iith.ac.in
Weiyong Ding
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 22116, China. E-mail: mathdwy@hotmail.com, zhaop@jsnu.edu.cn
Peng Zhao
Affiliation:
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 22116, China. E-mail: mathdwy@hotmail.com, zhaop@jsnu.edu.cn
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Abstract

Pareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$, $a > 0$. As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

Keywords

Convex transform orderDispersive orderParallel systemPareto distributionStar order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62N05: Reliability and life testing 62G30: Order statistics; empirical distribution functions 62D05: Sampling theory, sample surveys
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 950 - 962
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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