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 $p\geq \tilde{p}$ , 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. Copyright Springer Science+Business Media, LLC 2014