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Duke Economics Working Paper #95-03

Logconcavity versus Logconvexity: A Complete Characterization

Mark Yuying An

Abstract

This paper studies aspects of the broad class of log-concave probability distributions} that arise in the economics of uncertainty and information. Useful properties of univariate log-concave distributions are proven without imposing differentiability of density functions. We also discuss discrete and multivariate distributions. We propose simple non-parametric testing procedures for log-concavity. The test statistics are constructed to test one of the two implications of log-concavity: increasing hazard rate or {\em new-is-better-than-used} (NBU) property. The tests for increasing hazard rate are based on sample information of the normalized spacing of the order statistics. The tests for NBU property fall into the category of Hoeffding's U-statistics. The test procedures are illustrated with well known economic data where log-concavity is usually assumed.

JEL: D80, C12, C14

Published in Journal of Economic Theory, Vol. 80, No. 2, June 1998, pp. 350-369.