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Summary:
Let two random vectors $X_1$ and $X_2$ be jointly distributed as a normal distribution with mean $\mu$ and covariance matrix $\sum_1$. Let $\Pi (\lambda)$be the probability that $X_1\in C_1, X_2\in C_2$, where $C_1$ and $C_2$ are convex symmetric sets, when the covariance matrix between $X_1$ and $X_2$ is multiplied by $\lambda;0\leq \lambda \leq 1$. It is shown that $\Pi(\lambda)$ increases with $\lambda$ under some conditions on $\mu$ and $\sum_1$. This generalizes the results of Das Gupta et al (1972), Khatri (1967) and Šidák (1973).
References:
[1] Das Gupta S., Eaton M. L., Olkin I., Perlman M. D., Savage L. J., and Sobel M.: Inequalities on the probability content of convex regions for elliptically contoured distributions. Proc. Sixth Berk. Symp. on Math. Stat. and Prob. Vol. 11, (1972), 241-265. MR 0413364
[2] Khatri C. G.: On certain inequalities for normal distributions and their applications to simultaneous confidence bounds. Ann. Math. Statist. 38, (1967), 1853-1867. DOI 10.1214/aoms/1177698618 | MR 0220392 | Zbl 0155.27103
[3] Šidák Z.: On probabilities in certain multivariate distributions: their dependence on correlations. Aplikace Matematiky 18, (1973), 128-135. MR 0314197
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