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 X_i?X*_i] having shape parameter r_i?r*_i] and scale parameter λ_i?λ*_i], i = 1, 2. It is shown that the likelihood ratio order holds between the maxima, X_{2: 2} and X*_{2: 2} when λ₁ = λ*₁ ? λ₂ = λ*₂ and r₁ ? r*₁ ? r₂ = r*₂. We also prove that, if r_i, r*_i ∈ (0, 1], (r₁, r₂) majorizes (r*₁, r₂*), and (λ₁, λ₂) is p-larger than (λ*₁, λ₂*), then X_{2: 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.