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

##### MSC:
 62L12 Sequential estimation 62L15 Optimal stopping in statistics
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