# Comparisons for series and parallel systems with discrete Weibull components via separate comparisons of parameters

Document Type: Original Scientific Paper

Author

Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran

Abstract

‎In this paper‎, ‎we obtain the usual stochastic order of series and parallel systems comprising heterogeneous discrete Weibull (DW) components‎. ‎Suppose X1,...,Xn and Y1,...,Yn denote the independent component¢s lifetimes of two systems such that Xi ~ DW(bi‎, ‎pi) and Yi ~ DW(b*i‎, ‎p*i), i=1,...,n. We obtain the usual stochastic order between series systems‎ ‎when the vector \boldsymbolb is switched to the vector b*with respect to the majorization order‎, ‎and when the vector log (1-p) is switched to the vector log (1-p*) in the sense of the weak supermajorization order‎. ‎We also discuss the usual stochastic order between series systems by using the unordered majorization between the vectors log(1-p) and log (1-p*), and the p-majorization order between the parameters \boldsymbolb and b*. It is also shown that the usual stochastic order between parallel systems comprising heterogeneous discrete Weibull components when the vector log p is switched to the vector log p*in the sense of the weak supermajorization order‎. ‎These results enable us to find some lower bounds for the survival functions of a series and parallel systems consisting of independent heterogeneous discrete Weibull components.

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