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

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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 = mini∈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 = maxi∈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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Authors and Affiliations

  1. Facultad de Matematicas, Universidad de Murcia, 30100, Murcia, Spain

    Jorge Navarro

  2. Department of Mathematics, University of Arizona, Tucson, AZ, 85718, USA

    Moshe Shaked

Authors
  1. Jorge Navarro
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  2. Moshe Shaked
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Correspondence to Jorge Navarro.

Additional information

The work of J. Navarro was partially supported by Ministerio de Ciencia y Tecnologí a under grant BFM2003-02947 and Fundación Séneca under grant 00698/PI/04.

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Navarro, J., Shaked, M. Some properties of the minimum and the maximum of random variables with joint logconcave distributions. Metrika 71, 313–317 (2010). https://doi.org/10.1007/s00184-009-0232-9

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  • DOI: https://doi.org/10.1007/s00184-009-0232-9

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