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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| Institution: | 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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| Keywords: | |
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