A sequential procedure with asymptotically negative regret for estimating a normal mean (Q1192992)

Let \(X_ 1,X_ 2,\dots\) be independent and identically distributed normal random variables with unknown mean \(\mu\) and unknown variance \(\sigma^ 2>0\). The estimator of \(\mu\) is the sample mean \(\overline{X}_ n\) and the loss function is \(L_ n=A(\overline{X}_ n- \mu)^ 2+n\), \(A>0\). The proposed sequential procedure for estimating the mean is such that the difference between the corresponding risk and minimum fixed size risk is negative at \(\mu=0\) and \(1/2\) at \(\mu\neq 0\) asymptotically [cf. \textit{M. Woodroofe}, Ann. Stat. 5, 984-995 (1977; Zbl 0374.62081)].