Resumen
Let X1,…,Xn be independent exponential random variables with Xi having hazard rate ?i, i=1,…,n, and Y1,…,Yn be another independent random sample from an exponential distribution with common hazard rate ?. The purpose of this paper is to examine the mean residual life order between the second order statistics X2:n and Y2:n from these two sets of variables. It is proved that X2:n is larger than Y2:n in terms of the mean residual life order if and only if View the MathML source where View the MathML source and ?i=?-?i. It is also shown that X2:n is smaller than Y2:n in terms of the mean residual life order if and only if View the MathML source These results extend the corresponding ones based on hazard rate order and likelihood ratio order established by Pa?lta?nea [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.

Resumen
Generalized order statistics introduced by Kamps [1995. A concept of generalized order statistics. J. Statist. Plann. Inference 48, 1–23] provides a unified approach to a variety of models concerning ordered random variables. This paper carries out stochastic comparisons of conditional generalized order statistics in terms of the likelihood ratio order and the hazard rate order from two samples, and establishes some stochastic monotonicity properties. The main results strengthen and generalize the corresponding results established recently in the literature. Finally, some applications of the main results are given as well.