On the maxima of heterogeneous gamma variables

Yiying Zhang and Peng Zhao

Communications in Statistics - Theory and Methods, 2017, vol. 46, issue 10, 5056-5071

Abstract: 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.

Date: 2017
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DOI: 10.1080/03610926.2015.1091084

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