The Annals of Statistics

A Complete Class Theorem for Statistical Problems with Finite Sample Spaces

Lawrence D. Brown

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This paper contains a complete class theorem (Theorem 3.2) which applies to most statistical estimation problems having a finite sample space. This theorem also applies to many other statistical problems with finite sample spaces. The description of this complete class involves a stepwise algorithm. At each step of the process it is necessary to construct the Bayes procedures in a suitably modified version of the original problem. The complete class is a minimal complete class if the loss function is strictly convex. Some examples are given to illustrate the application of this complete class theorem. Among these is a new result concerning the estimation of the parameters of a multinomial distribution under a normalized quadratic loss function. (See Example 4.5).

Article information

Ann. Statist., Volume 9, Number 6 (1981), 1289-1300.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62C07: Complete class results
Secondary: 62C15: Admissibility 62F10: Point estimation 62F11 62C10: Bayesian problems; characterization of Bayes procedures

Complete class theorem finite sample space admissible procedures Bayes procedure estimation binomial distribution multinomial distribution strictly convex loss squared error loss maximum likelihood estimate


Brown, Lawrence D. A Complete Class Theorem for Statistical Problems with Finite Sample Spaces. Ann. Statist. 9 (1981), no. 6, 1289--1300. doi:10.1214/aos/1176345645.

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