CONFIDENCE REGIONS FOR MEANS OF MULTIVARIATE

NORMAL DISTRIBUTIONS AND A NON-SYMMETRIC CORRELATION INEQUALITY FOR GAUSSIAN MEASURE

STANISLAW J. SZAREK ELISABETH WERNER

Abstract. Let μ be a Gaussian measure (say, on R^{n}) and let K, L ⊆ R^{n} be such that K is convex, L is a “layer” (i.e. L = {x : a ≤ ⟨x, u⟩ ≤ b} for some a, b ∈ R and u ∈ R^{n}) and the centers of mass (with respect to μ) of K and L coincide. Then μ(K ∩ L) ≥ μ(K) · μ(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: Φ(x) > 1 − \frac{(8/π)^{\frac{1}{2}}}{3x+(x^{2}+8)^{\frac{1}{2}}} e^{−x^{2}/2} (valid for x > −1).