Reduction of variance for Gaussian densities via restriction to convex sets

Marek Kanter and Harold Proppe

Journal of Multivariate Analysis, 1977, vol. 7, issue 1, 74-81

Abstract: Let X be a random vector with values in n and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on n. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a [set membership, variant] n. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.

Keywords: Normal; density; covariance; matrix; logarithmically; concave; functions; convex; subsets; of; n (search for similar items in EconPapers)
Date: 1977
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