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October, 1968 On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations
Zbynek Sidak
Ann. Math. Statist. 39(5): 1425-1434 (October, 1968). DOI: 10.1214/aoms/1177698122

Abstract

For a random vector $(X_1,\cdots, X_k)$ having a $k$-variate normal distribution with zero mean values, Slepian [16] has proved that the probability $P\{X_1 < c_1,\cdots, X_k < c_k\}$ is a non-decreasing function of correlations. The present paper deals with the "two-sided" analogue of this problem, namely, if also the probability $P\{|X_1| < c_1,\cdots, |X_k| < c_k\}$ is a non-decreasing function of correlations. It is shown that this is true in the important special case where the correlations are of the form $\lambda_i\lambda_j\rho_{ij}, \{\rho_{ij}\}$ being some fixed correlation matrix (Section 1), and that it is true locally in the case of equicorrelated variables (Section 3). However, some counterexamples are offered showing that a complete analogue of Slepian's result does not hold in general (Section 4). Some applications of the main positive result are mentioned briefly (Section 2).

Citation

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Zbynek Sidak. "On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations." Ann. Math. Statist. 39 (5) 1425 - 1434, October, 1968. https://doi.org/10.1214/aoms/1177698122

Information

Published: October, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0169.50102
MathSciNet: MR230403
Digital Object Identifier: 10.1214/aoms/1177698122

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 5 • October, 1968
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