Abstract
It is shown that if (X1, X2, . . . , Xn) is a random vector with a logconcave (logconvex) joint reliability function, then XP = mini∈PXi has increasing (decreasing) hazard rate. Analogously, it is shown that if (X1, X2, . . . , Xn) has a logconcave (logconvex) joint distribution function, then XP = maxi∈PXi has decreasing (increasing) reversed hazard rate. If the random vector is absolutely continuous with a logconcave density function, then it has a logconcave reliability and distribution functions and hence we obtain a result given by Hu and Li (Metrika 65:325–330, 2007). It is also shown that if (X1, X2, . . . , Xn) has an exchangeable logconcave density function then both XP and XP have increasing likelihood ratio.
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