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 X 1 , … , X n be independent gamma random variables with X i having shape parameter r ∈ ( 0 , 1 ] and scale parameter λ i , i = 1 , … , n , and let X n : n denote the corresponding largest order statistic. Let Y n : n denote the largest order statistic arising from independent and identically distributed gamma random variables Y 1 , … , Y n with Y i having shape parameter r and scale parameter λ ̄ = ∑ i = 1 n λ i / n , the arithmetic mean of λ i ’s. It is shown here that X n : n is stochastically greater than Y n : 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.