论文标题
顺序假设检验的二阶渐近学
Second-Order Asymptotics of Sequential Hypothesis Testing
论文作者
论文摘要
我们考虑经典的顺序二进制假设检验问题，其中有两个假设分别由分布$ p_0 $和$ p_1 $控制，我们想决定使用顺序测试确定哪些假设是正确的。从沃尔德（Wald）和沃尔夫维茨（Wolfowitz）的工作中知道，随着测试长度的期望，最佳I型I和类型II错误指数接近相对熵$ d（p_1 \ | p_0）$和$ d（p_0 \ | p_0 \ | p_1）$。我们通过从可实现的指数区域$（d（p_1 \ | p_0），d（p_0 \ | p_1））$考虑最佳的退缩 - 或二阶渐近学---在测试长度（或样本大小）的两个不同约束下，我们可以完善该结果。首先，我们考虑一个概率约束，其中测试长度超过规定的整数$ n $的概率小于一定阈值$ 0 <\ varepsilon <1 $。其次，样本量的期望是$ n $的。在这两种情况下，在轻度条件下，二阶渐近肌均准确表征。提供数值示例以说明我们的结果。
We consider the classical sequential binary hypothesis testing problem in which there are two hypotheses governed respectively by distributions $P_0$ and $P_1$ and we would like to decide which hypothesis is true using a sequential test. It is known from the work of Wald and Wolfowitz that as the expectation of the length of the test grows, the optimal type-I and type-II error exponents approach the relative entropies $D(P_1\|P_0)$ and $D(P_0\|P_1)$. We refine this result by considering the optimal backoff---or second-order asymptotics---from the corner point of the achievable exponent region $(D(P_1\|P_0),D(P_0\|P_1))$ under two different constraints on the length of the test (or the sample size). First, we consider a probabilistic constraint in which the probability that the length of test exceeds a prescribed integer $n$ is less than a certain threshold $0<\varepsilon <1$. Second, the expectation of the sample size is bounded by $n$. In both cases, and under mild conditions, the second-order asymptotics is characterized exactly. Numerical examples are provided to illustrate our results.
