Exact lower and upper bounds for shifts of Gaussian measures

TL;DR

This paper derives exact bounds for ratios and rate of change of expectations involving shifted Gaussian vectors, with applications to statistical test power functions in multivariate normal settings.

Contribution

It provides the first precise bounds on Gaussian measure shifts and their implications for statistical test power analysis.

Findings

Exact bounds on Gaussian measure ratios and shifts.

Bounds on the rate of change of expectations under shifts.

Applications to the power function of multivariate normal tests.

Abstract

Exact upper and lower bounds on the ratio $\mathsf{E}w(\mathbf{X}-\mathbf{v})/\mathsf{E}w(\mathbf{X})$ for a centered Gaussian random vector $\mathbf{X}$ in $\mathbb{R}^n$, as well as bounds on the rate of change of $\mathsf{E}w(\mathbf{X}-t\mathbf{v})$ in $t$, where $w\colon\mathbb{R}^n\to[0,\infty)$ is any even unimodal function and $\mathbf{v}$ is any vector in $\mathbb{R}^n$. As a corollary of such results, exact upper and lower bounds on the power function of statistical tests for the mean of a multivariate normal distribution are given.