Stochastic Comparisons of Series and Parallel Systems With Generalized Exponential Components

Abstract

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

Stochastic Comparisons of Extreme Order Statistics in the Heterogeneous Exponentiated Scale Model

“…So the result of this Theorems simply yield results for series and parallel systems with independent heterogeneous generalized exponential components. As a consequence, this results is a generalization of Corresponding result due to [1].…”

“…From Theorem 3.6 and the fact that F is DPRHR and x 2r (x) is increasing in x, see lemma 2.1 of [12], we readily obtain the following corollary that generalizes the corresponding result in Theorem 10 (ii) of [1]. In particular the majorization assumption is relaxed to the super-majorization.…”

Ordering properties of the smallest and largest claim amounts in a general scale model

“…We say that two T -transform matrices T 1 = w 1 I n + (1 − w 1 ) 1 and T 2 = w 2 I n + (1 − w 2 ) 2 have the same structure if 1 = 2 . It is well known that the finite product of T -transform matrices with the same structure is also a T -transform matrix, while this product may not be a T -transform matrix if its elements do not have the same structure; see Balakrishnan et al (2015) for more details.…”

Stochastic comparison of aggregate claim amounts between two heterogeneous portfolios and its applications

“…For two vectors a = (a 1 , · · · , a n ) and b = (b 1 , · · · , b n ), let {a (1) , · · · , a (n) } and {b (1) , · · · , b (n) } denote the increasing arrangements of their components, respectively. Then, vector a is said to majorize vector…”