Stochastic order of sample range from heterogeneous exponential random variables

  • Authors:

  • Peng Zhao;Xiaohu Li

  • Affiliations:

  • School of mathematics and statistics, lanzhou university, lanzhou 730000, china e-mail: xhli@lzu.edu.cn;School of mathematics and statistics, lanzhou university, lanzhou 730000, china e-mail: xhli@lzu.edu.cn

  • Venue:
  • Probability in the Engineering and Informational Sciences
  • Year:
  • 2009

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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:nX1:n is proved to be larger than Yn:nY1: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.