Likelihood ratio order of parallel systems with heterogeneous Weibull components

Abstract

In this paper, we compare the largest order statistics arising from independent heterogeneous Weibull random variables based on the likelihood ratio order. Let $X_{1},\ldots ,X_{n}$ be independent Weibull random variables with $X_{i}$ having shape parameter $0<\alpha \le 1$ and scale parameter $\lambda _{i}$, $i=1,\ldots ,n$, and $Y_{1},\ldots ,Y_{n}$ be a random sample of size n from a Weibull distribution with shape parameter $0<\alpha \le 1$ and a common scale parameter $\overline{\lambda }=\frac{1}{n}\sum \nolimits _{i=1}^{n}\lambda _{i}$, the arithmetic mean of $\lambda _{i}^{'}s$. Let $X_{n:n}$ and $Y_{n:n}$ denote the corresponding largest order statistics, respectively. We then prove that $X_{n:n}$ is stochastically larger than $Y_{n:n}$ in terms of the likelihood ratio order, and provide numerical examples to illustrate the results established here.