Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Abstract

Let X λ 1 , X λ 2 , . . . , X λn be independent Weibull random variables with X λ i ∼ W (α, λi) where λi > 0, for i = 1, . . . , n. Let X λ n:n denote the lifetime of the parallel system formed from X λ 1 , X λ 2 , . . . , X λn . We investigate the effect of the changes in the scale parameters (λ1, . . . , λn) on the magnitude of X λ n:n according to reverse hazard rate and likelihood ratio orderings.

Stochastic comparison of parallel systems with heterogeneous exponential components

“…which implies that X n : n (λ) > mrl X n : n (μ). Second, we believe the following inequality holds: Finally, similar to [7], the result in this paper can be readily extended to parallel systems whose components follow proportional hazard models. That is, the survival function of X i satisfiesF X i (t) = e λ i R(t) , where R(t) is a baseline cumulative hazard function.…”

Stochastic comparisons of series and parallel systems with independent heterogeneous Gumbel and truncated Gumbel components

“…That is why series and parallel systems have been extensively studied for various lifetime distributions in the literature. Stochastic comparisons of the series and parallel systems with Weibull components have been considered recently by Li and Li (2015), Torrado (2015a), Torrado and Kochar (2015), Balakrishnan and Torrado (2016), Zhao et al (2016), Balakrishnan et al (2018a) and Wang (2018). Also the extended version of Weibull distribution has been studied for these systems by Fang and Zhang (2015), Fang and Balakrishnan (2016), Barmalzan et al (2017, 2019) and Balakrishnan et al (2018b).…”

Stochastic Comparisons of Parallel and Series Systems with Type II Half Logistic-Resilience Scale Components

“…However, the exponential distribution has the special feature of a constant failure rate, and it is not universally researched. Based on this, many scholars have generalized the exponential distribution to the Weibull distribution and gamma distribution (see the works of Fang and Zhang [6], Balakrishnan and Zhao [7], Zhao and Balakrishnan [8], Kochar and Torrado [9], Zhao et al [10], Torrado and Kochar [11], Zhao et al [12], and Zhang and Zhao [13], among others). In addition, there are some literature works that study the generalized exponential distribution, exponential Weibull distribution, and exponential generalized gamma distribution.…”