On the asymptotic of likelihood ratios for self-normalized large deviations

TL;DR

This paper derives the asymptotic behavior of likelihood ratios for self-normalized large deviations, providing insights into multiple testing error control when signals differ slightly from noise.

Contribution

It introduces a limit for likelihood ratios of self-normalized large deviations and applies it to determine minimum sample sizes for error control in multiple testing.

Findings

Limit of likelihood ratio is e^{d/z_0} for large n.

The result simplifies the analysis of error rates in multiple testing.

Application to small signal detection in noisy environments.

Abstract

Motivated by multiple statistical hypothesis testing, we obtain the limit of likelihood ratio of large deviations for self-normalized random variables, specifically, the ratio of $P(\sqrt{n}(\bar X +d/n) \ge x_n V)$ to $P(\sqrt{n}\bar X \ge x_n V)$, as $n\toi$, where $\bar X$ and $V$ are the sample mean and standard deviation of iid $X_1, ..., X_n$, respectively, $d>0$ is a constant and $x_n \toi$. We show that the limit can have a simple form $e^{d/z_0}$, where $z_0$ is the unique maximizer of $z f(x)$ with $f$ the density of $X_i$. The result is applied to derive the minimum sample size per test in order to control the error rate of multiple testing at a target level, when real signals are different from noise signals only by a small shift.