This paper examines the problem of the stochastic comparison of series and parallel systems with -independent heterogeneous generalized exponential components. The results established here are developed in three directions. First, we consider a system with possibly different shape and scale parameters, and obtain some ordering results when its matrix of parameters changes to another matrix, in the certain mathematical sense. Next, by using the concept of vector majorization and related orders, we establish various ordering results for the comparisons of series and parallel systems, when their component's lifetimes have either the same shape parameters with possibly different scale parameters, or the same scale parameters with possibly different shape parameters. Finally, some of the known results on various stochastic orderings between parallel systems in the exponential case are extended to the case when the lifetimes of components follow the generalized exponential distributions. The results of this paper can be used in practical situations to replace components of series and parallel systems by new components, or to find various bounds for the important aging characteristics of these systems.Index Terms-Generalized exponential distribution, hazard rate order, likelihood ratio order, multiple-outlier model, multivariate majorization, order statistics, parallel system, series system. ACRONYMSAND ABBREVIATIONS EW exponentiated weibull GE generalized exponential IHRF increasing hazard rate function NBUE new better than used in expectation NOTATION order statistics of distribution function of survival function of quantile function of Manuscript density function of hazard rate function of reversed hazard rate function of -expectation of variance of increasing arrangements of the components of the vector increasing arrangements of the components of the vector identity matrix of order the increasing convex order the usual stochastic order the hazard rate order the reversed hazard rate order the likelihood ratio order the dispersive order the right-spread order the convex transform order the new better than used in expectation order the majorization order the weak supermajorization order the weak submajorization order the p-larger order the chain majorization order

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Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

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The main aim of this paper is to present two new results concerning the characterization of likelihood ratio and reversed hazard rate orders between largest order statistics from two sets of independent heterogeneous and homogeneous exponentiated generalized gamma distributed random variables. These characterization results complete and strengthen some previous ones in the literature.