Increasing failure rate and decreasing reversed hazard rate properties of the minimum and maximum of multivariate distributions with log-concave densities

Abstract

For a multivariate random vector X = (X1,...,Xn) with a log-concave density function, it is shown that the minimum min{X1,...,Xn} has an increasing failure rate, and the maximum max{X1,...,Xn} has a decreasing reversed hazard rate. As an immediate consequence, the result of Gupta and Gupta (in Metrika 53:39–49, 2001) on the multivariate normal distribution is obtained. One error in Gupta and Gupta method is also pointed out.