On Stochastic Comparisons of Largest Order Statistics in the Scale Model

Let Xλ1,Xλ2,...,Xλn$X_{\lambda _{1}},X_{\lambda _{2}},\ldots,X_{\lambda _{n}}$ be independent non negative random variables with Xλi∼F(λit)$X_{\lambda _{i}}\sim F(\lambda _{i}t)$, i = 1, …, n, where λi > 0, i = 1, …, n and F is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic Xλn: n is smaller than another one Xθn: n 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.