In this paper, we review some recent results on the stochastic comparison of order statistics and related statistics from independent and heterogeneous proportional hazard rates models, gamma variables, geometric variables, and negative binomial variables. We highlight the close connections that exist between some classical stochastic orders and majorization-type orders.

D2 dissection was associated with a significantly higher postoperative risk. A trend towards lower gastric cancer-related mortality was found following D2 dissection that did not include resection of the spleen or pancreas, but further long-term survival data are needed to determine whether there is a specific survival benefit after D2 dissection.

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a b s t r a c tIn this paper, we study convolutions of heterogeneous exponential random variables with respect to the mean residual life order. By introducing a new partial order (reciprocal majorization order), we prove that this order between two parameter vectors implies the mean residual life order between convolutions of two heterogeneous exponential samples. For the 2-dimensional case, it is shown that there exists a stronger equivalence. We discuss, in particular, the case when one convolution involves identically distributed variables, and show in this case that the mean residual life order is actually associated with the harmonic mean of parameters. Finally, we derive the ''best gamma bounds'' for the mean residual life function of any convolution of exponential distributions under this framework.

a b s t r a c tLet X 1 , . . . , X n be independent exponential random variables with respective hazard rates λ 1 , . . . , λ n , and let Y 1 , . . . , Y n be independent exponential random variables with common hazard rate λ. This paper proves that X 2:n , the second order statistic of X 1 , . . . , X n , is larger than Y 2:n , the second order statistic of Y 1 , . . . , Y n , in terms of the likelihood ratio order ifAlso, it is shown that X 2:n is smaller than Y 2:n in terms of the likelihood ratio order if and only ifThese results form nice extensions of those on the hazard rate order in Pǎltǎnea [E. Pǎltǎnea, On the comparison in hazard rate ordering of fail-safe systems,

In this paper, we study ordering properties of lifetimes of parallel systems with two independent heterogeneous exponential components in terms of the likelihood ratio order (reversed hazard rate order) and the hazard rate order (stochastic order). We establish, among others, that the weakly majorization order between two hazard rate vectors is equivalent to the likelihood ratio order (reversed hazard rate order) between lifetimes of two parallel systems, and that the p-larger order between two hazard rate vectors is equivalent to the hazard rate order (stochastic order) between lifetimes of two parallel systems. Moreover, we extend the results to the proportional hazard rate models. The results derived here strengthen and generalize some of the results known in the literature.

Likelihood ratio order Dispersive order Hazard rate order Star order Right spread order Mean residual life order Majorization p-larger order Reciprocal majorization order Gamma convolution a b s t r a c tConvolutions of independent random variables often arise in a natural way in many applied areas. In this paper, we study various stochastic orderings of convolutions of heterogeneous gamma random variables in terms of the majorization order [p-larger order, reciprocal majorization order] of parameter vectors and the likelihood ratio order [dispersive order, hazard rate order, star order, right spread order, mean residual life order] between convolutions of two heterogeneous gamma sets of variables wherein they have both differing scale parameters and differing shape parameters. The results established in this paper strengthen and generalize those known in the literature.