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

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 $zf(x)$ 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.

typos on pages 1, 3 and 8 of the same type: missing or extra \sqrt{n} in the expressions of probabilities of large deviations