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 λ i , i = 1 , … , n , and Y 1 , … , Y n be another independent random sample from an exponential distribution with common hazard rate λ . 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 λ ⩾ ( 2 n - 1 ) n ( n - 1 ) ∑ i = 1 n 1 Λ i - n - 1 Λ , where Λ = ∑ i = 1 n λ i and Λ i = Λ - λ i . 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 λ ⩽ min 1 ⩽ i ⩽ n Λ i n - 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.