In a routinely occurring estimation problem, the experimenter can often consider the interested parameters themselves as random variables with unknown prior distribution. Without knowledge of the exact prior distribution the Bayes estimator cannot be obtained. However, as long as independent repetitions of the experiment occur, the empirical Bayes approach can then be applied. A general strategy underlying the empirical Bayes estimator consists of finding the Bayes estimator in a form which can be estimated sequentially by using the past data. Such use of the data circumvents knowledge of the prior distribution.
Three different types of sampling distributions of cell counts of 2x2 contingency tables were considered. In the Independent Poisson Case, an empirical Bayes estimator for the cross-product ratio is presented. If the squared error loss function is used, this empirical Bayes estimator, ᾶ, has asymptotic risk only ε > 0 larger than the true Bayes risk.
For the Product Binomial and Multinomial situations, several empirical Bayes estimators for α are proposed. Basically, these 'empirical' Bayes estimators by-pass the prior distribution by estimating the marginal probabilities P(X₁₁,X₂₁,X₁₂,X₂₂) and P(X₁₁+1,X₂₁-1,X₁₂-1,X₂₂+1), where (X₁₁,X₂₁,X₁₂,X₂₂) is the set of current cell counts. Furthermore, because of the assumption of varying sample size(s), they will have asymptotic risk only ε > 0 away from the true Bayes risk if both the number of past experiences and the sample size(s) are sufficiently large.
Results of Monte Carlo simulation of empirical Bayes estimators are presented for the carefully selected prior distributions. Mean squared errors for these estimators and classical estimators were compared. The improvement of empirical Bayes over classical estimators was found to be dependent upon the prior means, the prior variances, the prior distribution of the parameters considered as random variables, and sample size(s). These conclusions are summarized, and tables are provided.
The empirical Bayes estimators of α start to show significant improvement over classical estimators for as few as only ten past experiences. In many instances, the improvement is something on the order of 15% with only ten past experiences and sample size(s) larger than twenty. However, for the cases where the prior variances are very large, the empirical Bayes estimator indicates neither better nor worse over the classical. Greater improvement is shown for more past experiences until around thirty when the improvement appears stabilized.
Finally, the other existing estimators for a which also take into account past experiences are discussed and compared to the corresponding empirical Bayes estimator. They were proposed respectively by Birch, Goodman, Mantel and Haenszel, Woolf, etc. The simulation study for comparisons indicate that empirical Bayes estimators outmatch them even with small prior variance. A test for deciding when empirical Bayes estimators of α should be used is also suggested and discussed. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/94541 |
| Date | January 1981 |
| Creators | Lee, Luen-Fure |
| Contributors | Statistics |
| Publisher | Virginia Polytechnic Institute and State University |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | v, 158 pages, 2 unnumbered leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 7985323 |
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