A sequential procedure with asymptotically negative regret for estimating a normal mean. (English) Zbl 0777.62082

Takada, Yoshikazu

Ann. Stat. 20, No. 1, 562-569 (1992).

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. M. Woodroofe, Ann. Stat. 5, 984-995 (1977; Zbl 0374.62081)].

Reviewer: K.Szajowski (Wrocław)

MSC:

62L12	Sequential estimation
62L15	Optimal stopping in statistics

Keywords:

asymptotically negative regret; normal mean; uniform integrability; uniform continuity in probability; Wald’s lemma; Anscombe’s theorem; unknown mean; unknown variance; sample mean; minimum fixed size risk

Citations:

Zbl 0374.62081