Extensions of Stein's Lemma for the Skew-Normal Distribution

Abstract

When two random variables have a bivariate normal distribution, Stein's lemma (Stein, 1973 Stein , C. M. ( 1973 ). Estimation of the mean of a multivariate normal distribution . Proc. Prague Symp. Asymptotic Statist. 345 – 381 . [Google Scholar] 1981 Stein , C. M. ( 1981 ). Estimation of the mean of a multivariate normal distribution . Ann. Statist. 9 : 1135 – 1151 . [Crossref], [Web of Science ®] [Google Scholar] ), provides, under certain regularity conditions, an expression for the covariance of the first variable with a function of the second. An extension of the lemma due to Liu ( 1994 Liu , J. S. ( 1994 ). Siegel's formula via Stein's identities . Statist. Probab. Lett. 21 : 247 – 251 . [Crossref], [Web of Science ®] [Google Scholar] ) as well as to Stein himself establishes an analogous result for a vector of variables which has a multivariate normal distribution. The extension leads in turn to a generalization of Siegel's ( 1993 Siegel , A. F. ( 1993 ). A surprising covariance involving the minimum of multivariate normal variables . J. Amer. Statist. Assoc. 88 : 77 – 80 . [Taylor & Francis Online], [Web of Science ®] [Google Scholar] ) formula for the covariance of an arbitrary element of a multivariate normal vector with its minimum element. This article describes extensions to Stein's lemma for the case when the vector of random variables has a multivariate skew-normal distribution. The corollaries to the main result include an extension to Siegel's formula. This article was motivated originally by the issue of portfolio selection in finance. Under multivariate normality, the implication of Stein's lemma is that all rational investors will select a portfolio which lies on Markowitz's mean-variance efficient frontier. A consequence of the extension to Stein's lemma is that under multivariate skew-normality, rational investors will select a portfolio which lies on a single mean-variance-skewness efficient hyper-surface.