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

1997/01/22 by Stanislaw J. Szarek, Stanisław J. Szarek, Szarek, Stanislaw J. +3 · 2 citations
Mathematics · #46B09 #52A20 #60D05 #60E15 #62H20 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration #Metric Geometry (math.MG) #Probability (math.PR) #Statistical Distribution Estimation and Applications #Statistical Methods and Inference #Statistics Theory (math.ST) #math.FA #math.MG #math.PR #math.ST #msc:46B09 #msc:52A20 #msc:60D05 #msc:60E15 #msc:62H20 #stat.TH

paper · pdf · doi:10.48550/arxiv.math/9701205

arxiv created 1997/01/22 · openalex publication_date 1997/01/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let μ be a Gaussian measure (say, on \bf Rⁿ) and let K, L ⊂ \bf Rⁿ be such that K is convex, L is a "layer" (i.e. L = \x : a ≤ < x,u > ≤ b \ for some a, b ∈ \bf R and u ∈ \bf 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 - \frac(8/π)^1/23x + (x² +8)^1/2 e^-x2/2 (valid for x > -1).

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