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

  • Authors:

  • Peng Zhao;Xiaohu Li;N. Balakrishnan

  • Affiliations:

  • School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China;Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2009

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Abstract

Let X"1,...,X"n be independent exponential random variables with respective hazard rates @l"1,...,@l"n, and let Y"1,...,Y"n be independent exponential random variables with common hazard rate @l. This paper proves that X"2":"n, the second order statistic of X"1,...,X"n, is larger than Y"2":"n, the second order statistic of Y"1,...,Y"n, in terms of the likelihood ratio order if and only if @l=12n-1(2@L"1+@L"3-@L"1@L"2@L"1^2-@L"2) with @L"k=@?"i"="1^n@l"i^k,k=1,2,3. Also, it is shown that X"2":"n is smaller than Y"2":"n in terms of the likelihood ratio order if and only if @l@?@?i=1n@l"i-max1@?i@?n@l"in-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].