Hazard rate comparison of parallel systems with heterogeneous gamma components

Abstract

a b s t r a c tWe compare the hazard rate functions of the largest order statistic arising from independent heterogeneous gamma random variables and that arising from i.i.d. gamma random variables. Specifically, let X 1 , . . . , X n be independent gamma random variables with X i having shape parameter 0 < r ≤ 1 and scale parameter λ i , i = 1, . . . , n. Denote by Y n:n the largest order statistic arising from i.i.d. gamma random variables Y 1 , . . . , Y n with Y i having shape parameter r and scale parameterλ = (  n i=1 λ i ) 1/n , the geometric mean of λ ′ i s. It is shown that X n:n is stochastically larger than Y n:n in terms of hazard rate order. The result derived here strengthens and generalizes some of the results known in the literature and leads to a sharp upper bound on the hazard rate function of the largest order statistic from heterogeneous gamma variables in terms of that of the largest order statistic from i.i.d. gamma variables. A numerical example is finally provided to illustrate the main result established here.

Ordering Properties of Order Statistics From Heterogeneous Populations: A Review With an Emphasis on Some Recent Developments

“…gamma sample. The following numerical example, due to Balakrishnan and Zhao [6], can be used to illustrate this fact.…”

“…Example 2.4: Let (X 1 , X 2 , X 3 ) be a vector of independent exponential variables with parameter vector (1,2,9), and (Y 1 , Y 2 , Y 3 ) be another vector of independent exponential variables with parameter vector (1,5,6). Then, it is clear that (1,2,9) m (1, 5, 6).…”

Stochastic Comparisons of Series and Parallel Systems With Generalized Exponential Components

“…We refer the readers to Dykstra et al [7], Khaledi and Kochar [16], Joo and Mi [15], and Zhao and Balakrishnan [34], [35] for detailed discussions on this topic. Some of these results have been extended to the case when the lifetimes of components follow Weibull (Khaledi and Kochar [18], Kochar and Xu [20], [21], and Fang and Zhang [8]), and gamma (Zhao and Balakrishnan [36], and Balakrishnan and Zhao [3]) distributions, while the general case is discussed by Misra and Misra [27], and Ding et al [6].…”

Some stochastic comparison results for series and parallel systems with heterogeneous Pareto type components

“…For two sets of Weibull random variables, where one set contains n independent but heterogeneous random variables and the other set contains independent but identical random variables, Fang and Zhang in [12] obtained sufficient conditions for dispersive ordering of the largest order statistics. Balakrishnan and Zhao in [4] proved some hazard rate comparison results of parallel systems associated with heterogeneous gamma distributed components. Fang and Zhang in [13] stochastically compared parallel systems with exponentiated-Weibull components in terms of the usual stochastic order, dispersive order and the likelihood ratio order.…”