A stochastic inequality for the largest order statistics from heterogeneous gamma variables

In this paper, we compare the largest order statistics arising from independent heterogeneous gamma random variables based on the likelihood ratio order. Let X1,…,XnX1,…,Xn be independent gamma random variables with XiXi having shape parameter r∈(0,1]r∈(0,1] and scale parameter λiλi, i=1,…,ni=1,…,n, and let Xn:nXn:n denote the corresponding largest order statistic. Let Yn:nYn:n denote the largest order statistic arising from independent and identically distributed gamma random variables Y1,…,YnY1,…,Yn with YiYi having shape parameter rr and scale parameter λ̄=∑i=1nλi/n, the arithmetic mean of λiλi’s. It is shown here that Xn:nXn:n is stochastically greater than Yn:nYn:n in terms of the likelihood ratio order. The result established here answers an open problem posed by Balakrishnan and Zhao (2013), and strengthens and generalizes some of the results known in the literature. Numerical examples are also provided to illustrate the main result established here.