Some properties of the minimum and the maximum of random variables with joint logconcave distributions |
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Authors: | Jorge Navarro Moshe Shaked |
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Affiliation: | 1. Facultad de Matematicas, Universidad de Murcia, 30100, Murcia, Spain 2. Department of Mathematics, University of Arizona, Tucson, AZ, 85718, USA
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Abstract: | 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 i∈P 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 i∈P 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. |
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