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On a probability inequality for multivariate normal distribution
Applications of Mathematics, Tome 21 (1976) no. 1, pp. 1-4
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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).
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).
DOI : 10.21136/AM.1976.103618
Classification : 26D15, 52A20, 60E05, 62H05, 62H99 
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Das Gupta, Somesh. On a probability inequality for multivariate normal distribution. Applications of Mathematics, Tome 21 (1976) no. 1, pp. 1-4. doi: 10.21136/AM.1976.103618

[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

[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 | MR | Zbl

[3] Šidák Z.: On probabilities in certain multivariate distributions: their dependence on correlations. Aplikace Matematiky 18, (1973), 128-135. | MR

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