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

Let X1,…,XnX1,…,Xn be independent exponential random variables with respective hazard rates λ1,…,λnλ1,…,λn, and let Y1,…,YnY1,…,Yn be independent exponential random variables with common hazard rate λλ. This paper proves that X2:nX2:n, the second order statistic of X1,…,XnX1,…,Xn, is larger than Y2:nY2:n, the second order statistic of Y1,…,YnY1,…,Yn, in terms of the likelihood ratio order if and only if λ≥12n−1(2Λ1+Λ3−Λ1Λ2Λ12−Λ2) with Λk=∑i=1nλik,k=1,2,3. Also, it is shown that X2:nX2:n is smaller than Y2:nY2:n in terms of the likelihood ratio order if and only if λ≤∑i=1nλi−max1≤i≤nλin−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].