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

Let X-1, ... , X-n be independent exponential random variables with respective hazard rates lambda(1), ... , lambda(n) and let Y-1, ... , Y-n, be independent exponential random variables with common hazard rate lambda. 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 lambda >= 1/2n-1 (2 Lambda(1) + Lambda(3) -Lambda(1)Lambda(2)/Lambda(2)(1)-Lambda(2)) with Lambda(k) = Sigma(n)(i=1), 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 [GRAPHICS] These results form nice extensions of those on the hazard rate order in Paltanea [E. Paltanea, On the comparison in hazard rate ordering of fail-safe systems, journal of Statistical Planning and Inference 138 (2008) 1993-1997]. (C) 2008 Elsevier Inc. All rights reserved.