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STOCHASTIC ORDER OF SAMPLE RANGE FROM HETEROGENEOUS EXPONENTIAL RANDOM VARIABLES

Published online by Cambridge University Press:  13 November 2008

Peng Zhao  and
Xiaohu Li
Peng Zhao
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China E-mail: xhli@lzu.edu.cn
Xiaohu Li
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China E-mail: xhli@lzu.edu.cn
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Abstract

Let X1, …, Xn be independent exponential random variables with their respective hazard rates λ1, …, λn, and let Y1, …, Yn be independent exponential random variables with common hazard rate λ. Denote by Xn:n, Yn:n and X1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n−X1:n is proved to be larger than Yn:n−Y1:n according to the usual stochastic order if and only if with . Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 23 , Issue 1 , January 2009 , pp. 17 - 29
Copyright
Copyright © Cambridge University Press 2009

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