Complete classes of two-stage estimation procedures for certain finite sample problems

Lee, Albert Fu-Yuan

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Permalink

https://hdl.handle.net/2142/20431

Description

Title

Complete classes of two-stage estimation procedures for certain finite sample problems

Author(s)

Lee, Albert Fu-Yuan

Issue Date

1990

Doctoral Committee Chair(s)

Marden, John I.

Department of Study

Statistics

Discipline

Statistics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Statistics

Language

eng

Abstract

Two-stage Bayes procedures, also known as Bayes double sample procedures, for estimating the mean of exponential family distributions are given by Cohen and Sackrowitz (1984). In their study, they develop double sample Bayes estimation procedures for the mean of exponential family distributions with respect to conjugate prior distributions. The procedures consist of stating $n\sb1$, the size of the first sample; $n\sb2$, the size of the second sample which depends on the data from the first sample; and finally the point estimate, which depends on the combined sample. The loss functions usually are linear combinations of loss due to terminal decision and loss due to sampling. They find the optimal second sample size as well as the optimal first sample size.

In our study, the first sample size is fixed and the second sample is determined by discrete search, which is different from their determination. The admissibility of a generalized Bayes procedure with respect to an improper prior is investigated using Blyth's limiting Bayes method (1952) and Brown's totally Bayes method (1981). A complete class for two-stage estimation of the parameter p of the binomial distribution is obtained. In the several binomial distributions situation, we find the two-stage Bayes procedures for estimating the difference of two binomial parameters with respect to two independent prior distributions. The admissibility of a generalized Bayes procedure with respect to two independent improper prior distributions is investigated using Blyth's method and Brown's method. An analogous complete class theorem is obtained. Finally, we look at the multinomial situation. The prior distribution is now Dirichlet with ($\alpha\sb1$,$\alpha\sb2$,$\...$,$\alpha\sb{k}$) and the loss function is $\sum\sbsp{i = 1}{k}$($\ p\sb{i} -\ p\sb{i})\sp2 +\ c(n\sb1 +\ n\sb2)$. We find the two-stage Bayes procedures with respect to Dirichlet prior when k = 3. The admissibility of the procedure when $\alpha\sb1$ = $\alpha\sb2$ = $\alpha\sb3$ = 0 is obtained by a totally Bayes procedure with respect to a sequence of five-prior distributions. We also find a complete class theorem.

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