On stochastic comparisons of largest order statistics in the scale model
Let $X_{λ_{1}},X_{λ_{2}},\ldots ,X_{λ_{n}}$ be independent nonnegative random variables with $X_{λ_{i}}\sim F(λ_{i}t)$, $i=1,\ldots ,n$, where $λ_{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{λ}$ is smaller than another one $X_{n:n}^{θ}$ 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.
math.ST↗