Characterization of MRL order of fail-safe systems with heterogeneous exponential components

Let X-1, ... , X-n be independent exponential random variables with X-i having hazard rate lambda(i), i = 1, ... , n, and Y-1, ... , Y-n be another independent random sample from an exponential distribution with common hazard rate lambda The purpose of this paper is to examine the mean residual life order between the second order statistics X-2:n and Y-2:n from these two sets of variables. It is proved that X-2:n is larger than Y-2:n in terms of the mean residual life order if and only if lambda >= (2n - 1)/n(n - 1) (Sigma(n)(i=1) 1/Lambda(i) - n - 1/Lambda) where Lambda = Sigma(n)(i=t) and Lambda(1) = Lambda - lambda(1). It is also shown that X-2:n is smaller than Y-2:n in terms of the mean residual life order if and only if lambda <= min(1 <= i <= n) Lambda(i)/n - 1 These results extend the corresponding ones based oil hazard rate order and likelihood ratio order established by Paltanea [2008. Oil the comparison in hazard rate ordering of fail-safe systems. journal of Statistical Planning and Inference 138, 1993-1997] and Zhao et al. [2009. Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. journal of Multivariate Analysis 100, 952-962], respectively. (C) 2009 Elsevier B.V. All rights reserved.