Stochastic Comparisons Of Parallel Systems Research Articles

Comparisons of Parallel Systems with Components Having Proportional Reversed Hazard Rates and Starting Devices

In this paper, we consider stochastic comparisons of parallel systems with proportional reversed hazard rate (PRHR) distributed components equipped with starting devices. By considering parallel systems with two components that PRHR and starting devices, we prove the hazard rate and reversed hazard rate orders. These results are then generalized for such parallel systems with n components in terms of usual stochastic order. The establish results are illustrated with some examples.

Stochastic comparisons of parallel systems with starting devices

In this paper, we consider stochastic comparisons of parallel systems with location-scale distributed components equipped with starting devices. We specifically establish the hazard rate order of parallel systems with two location-scale distributed components equipped with starting devices when the starting devices have different probabilities but with the same location-scale distributed component. Next, we present the usual stochastic order of parallel systems with n location-scale distributed components equipped with starting devices. We also provide some numerical examples to illustrate all the comparison results established here.

Stochastic comparisons of parallel systems with heterogeneous exponentiated half logistic-F components

This study endeavors to compare two parallel systems delineated via exponentiated half logistic-F distributions with both identical (F) and non-identical (F 1 and F 2) baseline distribution functions. The comparisons are carried out with respect to the usual stochastic order as well as the reversed hazard rate order through the majorized shape parameters of the distributions.

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

PurposeThe purpose of this paper is to investigate the stochastic comparisons of the parallel system with independent heterogeneous Gumbel components and series and parallel systems with independent heterogeneous truncated Gumbel components in terms of various stochastic orderings.Design/methodology/approachThe obtained results in this paper are obtained by using the vector majorization methods and results. First, the components of series and parallel systems are heterogeneous and having Gumbel or truncated Gumbel distributions. Second, multiple-outlier truncated Gumbel models are discussed for these systems. Then, the relationship between the systems having Gumbel components and Weibull components are considered. Finally, Monte Carlo simulations are performed to illustrate some obtained results.FindingsThe reversed hazard rate and likelihood ratio orderings are obtained for the parallel system of Gumbel components. Using these results, similar new results are derived for the series system of Weibull components. Stochastic comparisons for the series and parallel systems having truncated Gumbel components are established in terms of hazard rate, likelihood ratio and reversed hazard rate orderings. Some new results are also derived for the series and parallel systems of upper-truncated Weibull components.Originality/valueTo the best of our knowledge thus far, stochastic comparisons of series and parallel systems with Gumbel or truncated Gumble components have not been considered in the literature. Moreover, new results for Weibull and upper-truncated Weibull components are presented based on Gumbel case results.

Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices

<p style='text-indent:20px;'>In this paper, we discuss stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. To begin with, we present the hazard rate order of parallel systems with two scale proportional hazards components equipped with starting devices for two different cases: first when the starting devices with different probability have the same scale proportional hazards components, and the second when the different scale proportional hazards components have the same starting devices probability. Next, we present the usual stochastic order of parallel systems with <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> scale proportional hazards components equipped with starting devices. Finally, we provide some numerical examples to illustrate all the results established here.</p>

Stochastic comparisons of parallel systems with generalized Kumaraswamy-G components

This paper treats the problem of stochastic comparisons of two parallel systems with independent heterogeneous components having lifetimes following exponentiated Kumaraswamy-G model. The cases of same and different parent distribution functions are considered. Majorization type partial orders-based sufficient conditions in comparing the largest order statistics in terms of the usual stochastic order, reversed hazard rate order and likelihood ratio order are obtained. The likelihood ratio order among largest order statistics is established for the heterogeneous multiple-outlier exponentiated Kumaraswamy-G models. Several numerical examples are presented for illustrations as well.

Stochastic comparisons of largest-order statistics for proportional reversed hazard rate model and applications

AbstractWe investigate stochastic comparisons of parallel systems (corresponding to the largest-order statistics) with respect to the reversed hazard rate and likelihood ratio orders for the proportional reversed hazard rate (PRHR) model. As applications of the main results, we obtain the equivalent characterizations of stochastic comparisons with respect to the reversed hazard rate and likelihood rate orders for the exponentiated generalized gamma and exponentiated Pareto distributions. Our results recover and strengthen some recent results in the literature.

Stochastic comparison of parallel systems with log-Lindley distributed components

In this paper, we study stochastic comparisons of parallel systems having log-Lindley distributed components. These comparisons are carried out with respect to reversed hazard rate and likelihood ratio ordering.

Stochastic comparisons of parallel systems with exponentiated Weibull components

The exponentiated Weibull (EW) distribution is the first generalization of the two-parameter Weibull distribution to accommodate nonmonotone hazard rates, including the bathtub shaped hazard rate. In this paper, we discuss stochastic comparisons of parallel systems with exponentiated Weibull components in terms of the usual stochastic order, dispersive order and the likelihood ratio order. We give some sufficient conditions for stochastic comparisons between lifetimes of parallel systems.

Some new results on stochastic comparisons of parallel systems

Let X1,…,Xn be independent exponential random variables with Xi having hazard rate . Let Y1,…,Yn be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏i=1nλi)1/n, the geometric mean of the λis. Let Xn:n = max{X1,…,Xn}. It is shown that Xn:n is greater than Yn:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of Xn:n and an upper bound on the hazard rate function of Xn:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference65, 203–211), which are in terms of the arithmetic mean of the λis. Furthermore, let X1*,…,Xn∗ be another set of independent exponential random variables with Xi∗ having hazard rate λi∗, i = 1,…,n. It is proved that if (logλ1,…,logλn) weakly majorizes (logλ1∗,…,logλn∗, then Xn:n is stochastically greater than Xn:n∗.

Some new results on stochastic comparisons of parallel systems

Let X 1,…,X n be independent exponential random variables with X i having hazard rate . Let Y 1,…,Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i=1 n λ i )1/n , the geometric mean of the λis. Let X n:n = max{X 1,…,X n }. It is shown that X n:n is greater than Y n:n according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X n:n and an upper bound on the hazard rate function of X n:n in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference 65, 203–211), which are in terms of the arithmetic mean of the λ i s. Furthermore, let X 1 *,…,X n ∗ be another set of independent exponential random variables with X i ∗ having hazard rate λ i ∗, i = 1,…,n. It is proved that if (logλ1,…,logλ n ) weakly majorizes (logλ1 ∗,…,logλ n ∗, then X n:n is stochastically greater than X n:n ∗.

Stochastic comparisons of parallel systems of heterogeneous exponential components

Let X 1, …, X n be independent exponential random variables with X i having hazard rate λ i , i = 1, …, n. Let λ = ( λ 1, …, λ n ). Let Y 1, …, Y n be a random sample of size n from an exponential distribution with common hazard rate λ = Σ n i = 1 λ i n . The purpose of this paper is to study stochastic comparisons between the largest order statistics X n: n and Y n: n from these two samples. It is proved that the hazard rate of X n: n is smaller than that of Y n: n . This gives a convenient upper bound on the hazard rate of X n: n in terms of that of Y n: n . It is also proved that Y n: n is smaller than X n: n according to dispersive ordering. While it is known that the survival function of X n: n is Schur convex in λ, Boland, El-Neweihi and Proschan [ J. Appl. Prohab. 31 (1994) 180–192] have shown that for n > 2, the hazard rate of X n: n is not Schur concave. It is shown here that, however, the reversed hazard rate of X n: n is Schur convex in λ.