On maximum order statistics from heterogeneous geometric variables

Let X ₁,X ₂ be independent geometric random variables with parameters p ₁,p ₂, respectively, and Y ₁,Y ₂ be i.i.d. geometric random variables with common parameter p. It is shown that X _2:2, the maximum order statistic from X ₁,X ₂, is larger than Y _2:2, the second order statistic from Y ₁,Y ₂, in terms of the hazard rate order [usual stochastic order] if and only if $pgeq tilde{p}$ , where $tilde{p}=(p_{1}p_{2})^{frac{1}{2}}$ is the geometric mean of (p ₁,p ₂). This result answers an open problem proposed recently by Mao and Hu (Probab. Eng. Inf. Sci. 24:245–262, 2010) for the case when n=2.