Log-concavity of the extremes from Gumbel bivariate exponential distributions

Abstract

In the classical risk theory, it is often used that different type dimensions can be aggregated into a single-dimensional statistic, as well as the assumption of properties on log-concavity of this aggregation. The extreme-order statistics, minimum and maximum, might be used as aggregate statistics. In this paper, we discuss the log-concavity of the survival function of the minimum and maximum from Gumbel bivariate exponential models, through the log-concavity of generalized mixtures of four or fewer exponential distributions, extending the papers of Baggs and Nagaraja [Baggs, G.E. and Nagaraja, H.N., 1996, Reliability properties of order statistics from bivariate exponential distributions. Communications in Statistics—Stochastic Models, 12, 611–631] and Franco and Vivo [Franco, M. and Vivo, J.M., 2002, Reliability properties of series and parallel systems from bivariate exponential models. Communications in Statistics—Theory and Methods, 31, 2349–2360] devote to the log-concavity for generalized mixtures of three or fewer exponential distributions.

Keywords:

• Generalized mixtures
• Bivariate exponential
• Minimum
• Maximum
• Log-concavity

Acknowledgements

The authors are grateful to the referees for the helpful comments and remarks.