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Some new results on stochastic comparisons of parallel systems

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

Published online by Cambridge University Press:  14 July 2016

Baha-Eldin Khaledi  and
Subhash Kochar
Baha-Eldin Khaledi*
Affiliation:
Indian Statistical Institute
Subhash Kochar*
Affiliation:
Indian Statistical Institute
*
∗ Postal address: Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India.
∗ Postal address: Indian Statistical Institute, 7 S. J. S. Sansanwal Marg, New Delhi 110016, India.
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Abstract

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.

Keywords

Exponential distributionhazard rate orderingdispersive orderingSchur functionsmajorizationp-larger ordering

MSC classification

Primary: 60E15: Inequalities; stochastic orderings 62N05: Reliability and life testing 62G30: Order statistics; empirical distribution functions 62D05: Sampling theory, sample surveys
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 1123 - 1128
Copyright
Copyright © by the Applied Probability Trust 2000 

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