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Some new results on stochastic comparisons of parallel systems

Part of:Survival analysis and censored dataNonparametric inferenceDistribution theory - ProbabilitySampling theory, sample surveys

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 X 1,…,X n be independent exponential random variables with X i having hazard rate . Let Y 1,…,Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1 n λi )1/n , the geometric mean of the λis. Let X n:n = max{X 1,…,X n }. It is shown that X n:n is greater than Y n:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n:n and an upper bound on the hazard rate function of X n: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 λi s. Furthermore, let X 1 *,…,X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λi ∗, i = 1,…,n. It is proved that if (logλ1,…,logλn ) weakly majorizes (logλ1 ∗,…,logλn ∗, then X n:n is stochastically greater than X n:n ∗.

Keywords

Exponential distributionhazard rate orderingdispersive orderingSchur functionsmajorizationp-larger ordering

MSC classification

Primary:60E15: Inequalities; stochastic orderings62N05: Reliability and life testing62G30: Order statistics; empirical distribution functions62D05: Sampling theory, sample surveys

Information

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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