Asymptotic Expansions in Anscombe's Theorem for Repeated Significance Tests and Estimation after Sequential Testing

Abstract

Let $x_1, x_2, \cdots$ be independent and normally distributed with unknown mean $\theta$ and variance 1. Let $\tau = \inf \{n \geq 1: |s_n| \geq \sqrt{2a(n + c)}\}$. Then a repeated significance test for a normal mean rejects the hypothesis $\theta = 0$ if and only if $\tau \leq N_0$ for some positive integer $N_0$. The problem we consider is estimation of $\theta$ based on the data $x_1,\cdots, x_T, T = \min\{\tau, N_0\}$. We shall solve this problem by obtaining the asymptotic expansion of the distribution of $(s_\tau - \tau\theta)/\sqrt{\tau}$ as $a \rightarrow \infty$, and then constructing the confidence intervals for $\theta$.