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Estimation in the general linear model when the accuracy is specified before data collection. (English) Zbl 0589.62067

The author uses a concept of accuracy saying that an estimator \({\hat \beta}\) of \(\beta\) is ”accurate with accuracy A and confidence c”, \(0<c<1\), if P(\({\hat \beta}\)-\(\beta\in A)\geq c\) for all \(\beta\). Note that A need only be any Borel set having an interior point at zero.
Given a sequence \(Y_ 1,Y_ 2,..\). of independent vector-valued homoscedastic normally-distributed random variables generated via the general linear model \(Y_ i=X_ i\beta +\epsilon\), the k-dimensional parameter \(\beta\) is accurately estimated using a sequential version of the well-known maximum probability estimator developed by L. Weiss and J. Wolfowitz [Ann. Inst. Stat. Math. 19, 193-206 (1967; Zbl 0183.212); see also ”Maximum probability estimators and related topics.” (1974; Zbl 0297.62015)]. The procedure also generalizes C. Stein’s [Ann. Math. Stat. 16, 243-258 (1945; Zbl 0060.304)] fixed-width confidence sets to several dimensions. Some examples are given to illustrate the procedure.
Reviewer: V.Mammitzsch

MSC:

62L12 Sequential estimation
62E20 Asymptotic distribution theory in statistics
60G40 Stopping times; optimal stopping problems; gambling theory
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