Exact lower and upper bounds for shifts of Gaussian measures

05/19/2022

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by   Iosif Pinelis, et al.

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Exact upper and lower bounds on the ratio 𝖤w(𝐗-𝐯)/𝖤w(𝐗) for a centered Gaussian random vector 𝐗 in ℝ^n, as well as bounds on the rate of change of 𝖤w(𝐗-t𝐯) in t, where wℝ^n→[0,∞) is any even unimodal function and 𝐯 is any vector in 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.