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Extensions of Stein's lemma for the skew-normal distribution

Adcock, C.J. (2007) Extensions of Stein's lemma for the skew-normal distribution. Communications in Statistics - Theory and Methods, 36 (9-12). pp. 1661-1671. ISSN 0361-0926

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Abstract

When two random variables have a bivariate normal distribution, Stein's lemma (Stein, 1973, 1981), 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) 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) 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 skewnormality, rational investors will select a portfolio which lies on a single meanvariance-skewness efficient hyper-surface.

Item Type: Article
Copyright, Publisher and Additional Information: © 2007 Taylor & Francis. This is an author produced version of a paper subsequently published in Communications in Statistics - Theory and Methods. Uploaded in accordance with the publisher's self-archiving policy.
Keywords: hidden truncation models; multivariate skew-normal distribution; portfolio selection; Siegel's formula; Stein's lemma; utility functions
Academic Units: The University of Sheffield > Faculty of Social Sciences (Sheffield) > Sheffield University Management School
Depositing User: Miss Anthea Tucker
Date Deposited: 21 Jun 2010 15:38
Last Modified: 08 Feb 2013 17:00
Published Version: http://dx.doi.org/10.1080/03610920601126084
Status: Published
Publisher: Taylor & Francis
Identification Number: 10.1080/03610920601126084
URI: http://eprints.whiterose.ac.uk/id/eprint/10953

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