Geodesic
Exact lower and upper bounds for Gaussian measures
Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 3, pp. 607-617 Cet article a éte moissonné depuis la source Math-Net.Ru

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Exact upper and lower bounds on the ratio $\operatorname{\mathbf{E}}w(\mathbf{X}-\mathbf{v})/\operatorname{\mathbf{E}}w(\mathbf{X})$ for a centered Gaussian random vector $\mathbf{X}$ in $\mathbf{R}^n$ are obtained, as well as bounds on the rate of change of $\operatorname{\mathbf{E}}w(\mathbf{X}-t\mathbf{v})$ in $t$, where $w\colon\mathbf{R}^n\to[0,\infty)$ is any even unimodal function and $\mathbf{v}$ is any vector in $\mathbf{R}^n$. As a corollary of such results, exact upper and lower bounds on the power function of statistical tests for the mean of a multivariate normal distribution are given.
Keywords: Gaussian measures, multivariate normal distribution, shifts, unimodality, logconcavity, monotonicity, exact bounds, tests for the mean. 
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I. Pinelis. Exact lower and upper bounds for Gaussian measures. Teoriâ veroâtnostej i ee primeneniâ, Tome 67 (2022) no. 3, pp. 607-617. http://geodesic.mathdoc.fr/item/TVP_2022_67_3_a11/

[1] T. W. Anderson, “The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities”, Proc. Amer. Math. Soc., 6:2 (1955), 170–176 | DOI | MR | Zbl

[2] H. J. Brascamp, E. H. Lieb, “On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation”, J. Funct. Anal., 22:4 (1976), 366–389 | DOI | MR | Zbl

[3] G. B. Folland, Real analysis. Modern techniques and their applications, Pure Appl. Math. (N. Y.), John Wiley Sons, Inc., New York, 1984, xiv+350 pp. | MR | Zbl

[4] R. J. Gardner, “The Brunn–Minkowski inequality”, Bull. Amer. Math. Soc. (N.S.), 39:3 (2002), 355–405 | DOI | MR | Zbl

[5] A. W. Marshall, I. Olkin, “Majorization in multivariate distributions”, Ann. Statist., 2:6 (1974), 1189–1200 | DOI | MR | Zbl

[6] I. Pinelis, “On l'Hospital-type rules for monotonicity”, JIPAM. J. Inequal. Pure Appl. Math., 7:2 (2006), 40, 19 pp. | MR | Zbl

[7] I. Pinelis, “Schur${}^2$-concavity properties of Gaussian measures, with applications to hypotheses testing”, J. Multivariate Anal., 124 (2014), 384–397 | DOI | MR | Zbl

[8] R. T. Rockafellar, Convex analysis, Princeton Landmarks Math., Reprint of the 1970 original, Princeton Univ. Press, Princeton, NJ, 1997, xviii+451 pp. | MR | Zbl | Zbl