On the maxima of heterogeneous gamma variables

In this article, we establish some new results on stochastic comparisons of the maxima of two heterogenous gamma variables with different shape and scale parameters. Let X1 and X2 [X*1 and X*2] be two independent gamma variables with Xi [X*i] having shape parameter ri [r*i] and scale parameter λi [λ*i], i = 1, 2. It is shown that the likelihood ratio order holds between the maxima, X2: 2 and X*2: 2 when λ1 = λ*1 ⩽ λ2 = λ*2 and r1 ⩾ r*1 ⩾ r2 = r*2. We also prove that, if ri, r*i ∈ (0, 1], (r1, r2) majorizes (r*1, r2*), and (λ1, λ2) is p-larger than (λ*1, λ2*), then X2: 2 is larger than X*2: 2 in the sense of the hazard rate order [dispersive order]. Some numerical examples are provided to illustrate the main results. The new results established here strengthen and generalize some of the results known in the literature.