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

Abstract: Let be independent exponential random variables with respective hazard rates , and let be independent exponential random variables with common hazard rate . This paper proves that , the second order statistic of , is larger than , the second order statistic of , in terms of the likelihood ratio order if and only if with . Also, it is shown that is smaller than in terms of the likelihood ratio order if and only if 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]. [Copyright &y& Elsevier]