On the maxima of heterogeneous gamma variables with different shape and scale parameters

In this article, we study the stochastic properties of the maxima from two independent heterogeneous gamma random variables with different both shape parameters and scale parameters. Our main purpose is to address how the heterogeneity of a random sample of size 2 affects the magnitude, skewness and dispersion of the maxima in the sense of various stochastic orderings. Let $$X_{1}$$ X 1 and $$X_{2}$$ X 2 be two independent gamma random variables with $$X_{i}$$ X i having shape parameter $$r_{i}>0$$ r i > 0 and scale parameter $$\lambda _{i}$$ λ i , $$i=1,2$$ i = 1 , 2 , and let $$X^{*}_{1}$$ X 1 ∗ and $$X^{*}_{2}$$ X 2 ∗ be another set of independent gamma random variables with $$X^{*}_{i}$$ X i ∗ having shape parameter $$r_{i}^{*}>0$$ r i ∗ > 0 and scale parameter $$\lambda _{i}^{*}$$ λ i ∗ , $$i=1,2$$ i = 1 , 2 . Denote by $$X_{2:2}$$ X 2 : 2 and $$X^{*}_{2:2}$$ X 2 : 2 ∗ the corresponding maxima, respectively. It is proved that, among others, if $$(r_{1},r_{2})$$ ( r 1 , r 2 ) majorize $$(r_{1}^{*},r_{2}^{*})$$ ( r 1 ∗ , r 2 ∗ ) and $$(\lambda _{1},\lambda _{2})$$ ( λ 1 , λ 2 ) weakly majorize $$(\lambda _{1}^{*},\lambda _{2}^{*})$$ ( λ 1 ∗ , λ 2 ∗ ) , then $$X_{2:2}$$ X 2 : 2 is stochastically larger that $$X^{*}_{2:2}$$ X 2 : 2 ∗ in the sense of the likelihood ratio order. We also study the skewness according to the star order for which a very general sufficient condition is provided, using which some useful consequences can be obtained. The new results established here strengthen and generalize some of the results known in the literature. Copyright Springer-Verlag Berlin Heidelberg 2014