标题： 关于自归一化大偏差的似然比的渐近性 显示英文标题

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

摘要： 受多重统计假设检验的启发，我们得到了自归一化随机变量的大偏差似然比的极限，具体来说，当 $n\toi$ 时，该比值为 $P(\sqrt{n}(\bar X +d/n) \ge x_n V)$ 与 $P(\sqrt{n}\bar X \ge x_n V)$ 的比值，其中 $\bar X$ 和 $V$ 分别是独立同分布 (iid) 随机变量 $X_1, ..., X_n$ 的样本均值和标准差，$d>0$ 是一个常数且 $x_n \toi$。 我们证明极限可以具有简单形式 $e^{d/z_0}$，其中 $z_0$ 是 $z f(x)$ 的唯一最大值点，而 $f$ 是 $X_i$ 的密度。 该结果被用于推导每次检验的最小样本量，以在真实信号仅比噪声信号大一个微小位移的情况下，将多重检验中的错误率控制在目标水平。 显示英文摘要

摘要： 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.