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September, 1950 Completeness in the Sequential Case
E. L. Lehmann, Charles Stein
Ann. Math. Statist. 21(3): 376-385 (September, 1950). DOI: 10.1214/aoms/1177729796


Recently, in a series of papers, Girshick, Mosteller, Savage and Wolfowitz have considered the uniqueness of unbiased estimates depending only on an appropriate sufficient statistic for sequential sampling schemes of binomial variables. A complete solution was obtained under the restriction to bounded estimates. This work, which has immediate consequences with respect to the existence of unbiased estimates with uniformly minimum variance, is extended here in two directions. A general necessary condition for uniqueness is found, and this is applied to obtain a complete solution of the uniqueness problem when the random variables have a Poisson or rectangular distribution. Necessary and sufficient conditions are also found in the binomial case without the restriction to bounded estimates. This permits the statement of a somewhat stronger optimum property for the estimates, and is applicable to the estimation of unbounded functions of the unknown probability.


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E. L. Lehmann. Charles Stein. "Completeness in the Sequential Case." Ann. Math. Statist. 21 (3) 376 - 385, September, 1950.


Published: September, 1950
First available in Project Euclid: 28 April 2007

zbMATH: 0041.26304
MathSciNet: MR36969
Digital Object Identifier: 10.1214/aoms/1177729796

Rights: Copyright © 1950 Institute of Mathematical Statistics

Vol.21 • No. 3 • September, 1950
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