Some properties of the minimum and the maximum of random variables with joint logconcave distributions

It is shown that if (X (1), X (2), . . . , X (n) ) is a random vector with a logconcave (logconvex) joint reliability function, then X (P) = min (iaP) X (i) has increasing (decreasing) hazard rate. Analogously, it is shown that if (X (1), X (2), . . . , X (n) ) has a logconcave (logconvex) joint distribution function, then X (P) = max (iaP) X (i) 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 (X (1), X (2), . . . , X (n) ) has an exchangeable logconcave density function then both X (P) and X (P) have increasing likelihood ratio.