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Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions

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Abstract

It is well known that in the case of independent random variables, the (reversed) hazard rate of the (maximum) minimum of two random variables is the sum of the individual (reversed) hazard rates and hence the onotonicity of the (reversed) hazard rate of the marginals is preserved by the monotonicity of the (reversed) hazard rate of the (maximum) minimum. However, for the bivariate distributions this property is not always preserved. In this paper, we study the monotonicity of the (reversed) hazard rate of the (maximum) minimum for two well known families of bivariate distributions viz the Farlie-Gumbel-Morgenstern (FGM) and Sarmanov family. In case of the FGM family, we obtain the (reversed) hazard rate of the (maximum) minimum and provide several examples in some of which the (reversed) hazard rate is monotonic and in others it is non-monotonic. In the case of Sarmanov family the (reversed) hazard rate of the (maximum) minimum may not be expressed in a compact form in general. We consider some examples to illustrate the procedure

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Maine, Orono, ME, 04469-5752, USA

    Ramesh C. Gupta & Pushpa L. Gupta

  2. Department of Computer Science and Applied Statistics, University of New Brunswick, Saint John, NB, Canada, E2L 4L5

    Rameshwar D. Gupta

Authors
  1. Ramesh C. Gupta
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  2. Rameshwar D. Gupta
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  3. Pushpa L. Gupta
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Correspondence to Ramesh C. Gupta.

Additional information

Research of the second author is supported by a grant from Natural Sciences and Engineering Research Council and the research of the other two authors is partially supported by a travel grant from the Canadian American Center of the University of Maine

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Gupta, R.C., Gupta, R.D. & Gupta, P.L. Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions. Metrika 63, 223–241 (2006). https://doi.org/10.1007/s00184-005-0014-y

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