The Annals of Mathematical Statistics

On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations

Zbynek Sidak

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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).

Article information

Source
Ann. Math. Statist., Volume 39, Number 5 (1968), 1425-1434.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177698122

Digital Object Identifier
doi:10.1214/aoms/1177698122

Mathematical Reviews number (MathSciNet)
MR230403

Zentralblatt MATH identifier
0169.50102

JSTOR
links.jstor.org

Citation

Sidak, Zbynek. On Multivariate Normal Probabilities of Rectangles: Their Dependence on Correlations. Ann. Math. Statist. 39 (1968), no. 5, 1425--1434. doi:10.1214/aoms/1177698122. https://projecteuclid.org/euclid.aoms/1177698122


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