On the expected sample sizes of some power one tests for normal mean with unknown variance (Q1092573)

Suppose that \(x_ 1,x_ 2,..\). are i.i.d. random variables on a probability space (\(\Omega\),\({\mathcal F},P_{\theta \sigma})\) and \(x_ 1\) is normally distributed with mean \(\theta\) and variance \(\sigma^ 2\), both of which are unknown. Given \(\theta_ 0\) and \(0<\alpha <1\), we propose a concrete stopping rule T w.r.t. the \(\{x_ n\), \(n\geq 1\}\) such that \[ P_{\theta \sigma}(T<\infty)\leq \alpha \text{ for all } \theta \leq \theta_ 0,\quad \sigma >0,\quad P_{\theta \sigma}(T<\infty)=1\text{ for all } \theta >\theta_ 0,\quad \sigma >0, \] \[ \lim_{\theta \downarrow \theta_ 0}(\theta -\theta_ 0)^ 2(\ln_ 2 1/(\theta -\theta_ 0))^{-1} E_{\theta \sigma}T=2\sigma^ 2P_{\theta_ 0\sigma}\quad (T=\infty), \] where \(\ln_ 2x=\ln (\ln x)\).