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arXiv 0709.1506math.STmath.PRstat.TH
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关于自归一化大偏差的似然比的渐近性

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

Zhiyi Chi

备注：
typos on pages 1, 3 and 8 of the same type: missing or extra \sqrt{n} in the expressions of probabilities of large deviations
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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的似然比的极限，具体地说，是$P(\sqrt{n}(\bar X +d/n) \ge x_n V)$与$P(\sqrt{n}\bar X \ge x_n V)$的比值，即$n\toi$，其中$\bar X$和$V$分别是iid $X_1, ..., X_n$的样本均值和标准差，$d>0$是常数，$x_n \toi$。我们证明了极限可以有一个简单的形式$e^{d/z_0}$，其中$z_0$是$z f(x)$的唯一极大化，$f$是$X_i$的密度。当真实信号与噪声信号仅相差一个小的偏移时，将结果应用于推导每次测试的最小样本量，以将多次测试的错误率控制在目标水平。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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