Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables

Peng Zhao, Xiaohu Li, N. Balakrishnan

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54 Scopus citations

Abstract

Let X1, ..., Xn be independent exponential random variables with respective hazard rates λ1, ..., λn, and let Y1, ..., Yn be independent exponential random variables with common hazard rate λ. This paper proves that X2 : n, the second order statistic of X1, ..., Xn, is larger than Y2 : n, the second order statistic of Y1, ..., Yn, in terms of the likelihood ratio order if and only if λ ≥ frac(1, 2 n - 1) (2 Λ1 + frac(Λ3 - Λ1 Λ2, Λ12 - Λ2)) with Λk = ∑i = 1n λik, k = 1, 2, 3. Also, it is shown that X2 : n is smaller than Y2 : n in terms of the likelihood ratio order if and only if λ ≤ frac(underover(∑, i = 1, n) λi - under(max, 1 ≤ i ≤ n) λi, n - 1) . These results form nice extensions of those on the hazard rate order in Pa ̌lta ̌nea [E. Pa ̌lta ̌nea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993-1997].

Original languageEnglish
Pages (from-to)952-962
Number of pages11
JournalJournal of Multivariate Analysis
Volume100
Issue number5
DOIs
StatePublished - May 2009

Keywords

  • 60E15
  • 60K10
  • Hazard rate order
  • Majorization order
  • Weakly majorization order
  • p-larger order
  • primary
  • secondary

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