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

الأصل · EN

Let μ be a Gaussian measure (say, on Rⁿ) and let K, L ⊂ Rⁿ 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 ∈ Rⁿ) 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 - (8/π)¹/²3x + (x² +8)¹/² e⁻ˣ²/² (valid for x > -1).