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Sufficient conditions for comparing the convolutions of heterogeneous gamma random variables in terms of the usual stochastic order are established. Such comparisons are characterized by the Schur convexity properties of the cumulative distribution function of the convolutions. Some examples of the practical applications of our results are given. [ABSTRACT FROM AUTHOR]

Professors Balakrishnan and Zhao have written an excellent survey on the recent developments of stochastic comparisons of order statistics, which cover almost every aspect of ordering properties of order statistics from both continuous and discrete heterogeneous populations. My discussion will be limited to the skewness of order statistics and order statistics from heterogeneous populations with different shape parameters. [ABSTRACT FROM AUTHOR]

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this paper generalize and unify several results in the literature including those of Amiri, Khaledi, and Samaniego [2], Zhao [18], and Kochar and Xu [9]. [ABSTRACT FROM AUTHOR]

In this article, mixture representations of survival functions of residual lifetimes of k-out-of-n systems are obtained when the components are independent but not necessarily identically distributed. Then we stochastically compare the residual lifetimes of k-out-of-n systems in one- and two-sample problems. In particular, the results extend some results in Li and Zhao [14], Khaledi and Shaked [13], Sadegh [17], Gurler and Bairamov [7] and Navarro, Balakrishnan, and Samaniego [16]. Applications in the proportional hazard rates model are presented as well. [ABSTRACT FROM AUTHOR]

First of all, we express our sincere thanks to all the discussants for their valuable comments and suggestions as well as for their own significant contributions to the area of order statistics in general, and to the topic of stochastic comparison in particular. We shall now provide our response to the comments and suggestions of all the discussants. [ABSTRACT FROM AUTHOR]