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

Abstract

Let X 1 , … , X n be independent exponential random variables with respective hazard rates λ 1 , … , λ n , and let Y 1 , … , Y n be independent exponential random variables with common hazard rate λ . 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 λ ≥ 1 2 n − 1 ( 2 Λ 1 + Λ 3 − Λ 1 Λ 2 Λ 1 2 − Λ 2 ) with Λ k = ∑ i = 1 n λ 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 λ ≤ ∑ i = 1 n λ i − max 1 ≤ i ≤ n λ i n − 1 . These results form nice extensions of those on the hazard rate order in Paˇltaˇnea [E. Paˇltaˇnea, On the comparison in hazard rate ordering of fail-safe systems, Journal of Statistical Planning and Inference 138 (2008) 1993–1997].