Likelihood ratio order of maxima from heterogeneous gamma random variables

In this paper, the largest order statistics arising from independent heterogeneous gamma random variables with respect to the likelihood ratio order are compared. Let X1, . . . , Xn (X∗ 1 , . . . , X∗ n ) be independent random variables where Xi (X∗ i ) follows the gamma distribution with shape parameter α and scale parameter λi(λ ∗ i ), in which α > 0, λi > 0 (λ ∗ i > 0), i = 1, . . . , n. Denote by Xn:n and X∗ n:n the corresponding largest order statistics, respectively. It is shown that, Xn:n is stochastically larger than X∗ n:n in terms of the likelihood ratio order if max{λ1, . . . , λn} ≤ min{λ ∗ 1 , . . . , λ∗ n}. The result derived here strengthens and generalizes some known results in the literature.