On stochastic comparisons of largest order statistics in the scale model

TL;DR

This paper investigates stochastic comparisons of the largest order statistics in scale models, establishing likelihood ratio orderings under certain conditions, with applications to generalized gamma distributions including Weibull, gamma, and exponential cases.

Contribution

It provides new conditions for likelihood ratio ordering of largest order statistics in scale models, extending to generalized gamma distributions.

Findings

Likelihood ratio ordering of largest order statistics established.

Results applicable to Weibull, gamma, and exponential distributions.

Conditions identified for stochastic comparisons in scale models.

Abstract

Let $X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$ be independent nonnegative random variables with $X_{\lambda _{i}}\sim F(\lambda _{i}t)$, $i=1,\ldots ,n$, where $\lambda _{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{\lambda }$ is smaller than another one $X_{n:n}^{\theta }$ according to likelihood ratio ordering. Furthermore, we apply these results when $F$ is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.