Confidence Regions for Means of Multivariate Normal Distributions and a Non-Symmetric Correlation Inequality for Gaussian Measure

Stanislaw J. Szarek and Elisabeth Werner

Let $\mu$ be a Gaussian measure (say, on ${\bf R}^n$) and let $K, L \subseteq {\bf R}^n$ be such that K is convex, $L$ is a ``layer" (i.e. $L = \{ x : a \leq \langle x,u \rangle \leq b \}$ for some $a$, $b \in {\bf R}$ and $u \in {\bf R}^n$) and the centers of mass (with respect to $\mu$) of $K$ and $L$ coincide. Then $\mu(K \cap L) \geq \mu(K) \cdot \mu(L)$. 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: $\Phi (x) > 1 - \frac{(8/\pi)^{\frac{1}{2}}}{3x + (x^2 +8)^{\frac{1}{2}}} e^{-x^2/2}$ (valid for $x > -1$).