Confidence regions for means of multivariate normal distributions and a non-symmetric correlation inequality for gaussian measure

Abstract

Let $\mu$ be a Gaussian measure (say, on ${\mathbf{R}}^n$) and let $K,L\subset {\mathbf{R}}^n$ be such that K is convex, $L$ is a "layer" (i.e. $L=\{x:a\le <x,u>\le b\}$ for some $a$, $b\in \mathbf{R}$ and $u\in {\mathbf{R}}^n$) and the centers of mass (with respect to $\mu$) of $K$ and $L$ coincide. Then $\mu (K\cap L)\ge \mu (K)\cdot \mu (L)$. This is motivated by the well-known "positive correlation conjecture" for symmetric sets and a related inequality of Sidak concerning confidence regions for means of multivariate normal distributions. The proof uses an apparently hitherto unknown estimate for the (standard) Gaussian cumulative distribution function: $\Phi (x)>1-\frac{(8/\pi {)}^{1/2}}{3x+(x^2+8{)}^{1/2}}{e}^{-x^2/2}$ (valid for $x>-1$).

References in corpus (1)

• On the Gaussian measure of the intersection of symmetric, convex sets

Cited by in corpus (3)

• Inequalities related to the error function
• One Dimensional Regularizations of the Coulomb Potential with Application to Atoms in Strong Magnetic Fields
• A pair of optimal inequalities related to the error function