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

Let X1,…,XnX1,…,Xn be independent exponential random variables with XiXi having hazard rate λiλi, i=1,…,ni=1,…,n, and Y1,…,YnY1,…,Yn be another independent random sample from an exponential distribution with common hazard rate λλ. The purpose of this paper is to examine the mean residual life order between the second order statistics X2:nX2:n and Y2:nY2:n from these two sets of variables. It is proved that X2:nX2:n is larger than Y2:nY2:n in terms of the mean residual life order if and only ifλ⩾(2n-1)n(n-1)∑i=1n1Λi-n-1Λ,where Λ=∑i=1nλi and Λi=Λ-λiΛi=Λ-λi. It is also shown that X2:nX2:n is smaller than Y2:nY2:n in terms of the mean residual life order if and only ifλ⩽min1⩽i⩽nΛin-1.These results extend the corresponding ones based on hazard rate order and likelihood ratio order established by Paˇltaˇnea [2008. On 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.