Some Relationships Between Skew-Normal Distributions and Order Statistics from Exchangeable Normal Random Vectors

Abstract

The article explores the relationship between distributions of order statistics from random vectors with exchangeable normal distributions and several skewed generalizations of the normal distribution. In particular, we show that the order statistics of exchangeable normal observations have closed skew-normal distributions, and that the corresponding density function is log-concave when the order statistic is extreme. Special attention is given to the bivariate case, which is related to the univariate skew-normal distribution. The applications discussed focus on the lifetimes of coherent systems.

Keywords:

• Closed skew-normal distribution
• Hazard rate
• Log-concavity
• Mean residual life
• Monotone likelihood ratio

Mathematics Subject Classification:

• Primary 60E15
• Secondary 60K10

Acknowledgments

We would like to thank Professor Azzalini for several helpful comments. We also thank the referees and guest editor Arthur Pewsey for thoughtful and useful feedback on previous versions of the article. Jorge Navarro and José M. Ruiz are partially supported by the Spanish Ministerio de Ciencia y Tecnología under grant BFM2003-02947 and the Fundación Séneca under grant 00698/PI/04.