Abstract
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.
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We would like to thank the editor and anonymous referees for careful reading of the manuscript and valuable comments and suggestions, which resulted in an improvement in the presentation of this manuscript.
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This work was supported by National Natural Science Foundation of China (11001112).
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Zhao, P., Su, F. On maximum order statistics from heterogeneous geometric variables. Ann Oper Res 212, 215–223 (2014). https://doi.org/10.1007/s10479-012-1158-6
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DOI: https://doi.org/10.1007/s10479-012-1158-6