A mean-squared error comparison of smooth empirical Bayes and Bayes estimators for the Weibull and gamma scale parameters is studied based on a computer simulation. The smooth empirical Bayes estimators are determined as functions of up to 15 past estimates of the parameter of interest. Results indi
Innovations in Bayes and empirical Bayes methods: estimating parameters, populations and ranks
β Scribed by Thomas A. Louis; Wei Shen
- Publisher
- John Wiley and Sons
- Year
- 1999
- Tongue
- English
- Weight
- 163 KB
- Volume
- 18
- Category
- Article
- ISSN
- 0277-6715
No coin nor oath required. For personal study only.
β¦ Synopsis
By formalizing the relation among components and &borrowing information' among them, Bayes and empirical Bayes methods can produce more valid, e$cient and informative statistical evaluations than those based on traditional methods. In addition, Bayesian structuring of complicated models and goals guides development of appropriate statistical approaches and generates summaries which properly account for sampling and modelling uncertainty. Computing innovations enable implementation of complex and relevant models, thereby substantially increasing the role of Bayes/empirical Bayes methods in important statistical assessments. Policy-relevant statistical assessments involve synthesis of information from a set of related components such as medical clinics, geographic regions or research studies. Typical assessments include inference for individual parameters, synthesis over the collection of components (for example, the parameter histogram) and comparisons among parameters (for example, ranks). The relative importance of these goals depends on the context. Bayesian structuring provides a guide to valid inference. For example, while posterior means are the &obvious' and optimal estimates for individual components under squared error loss, their empirical distribution function (EDF) is underdispersed and never valid for estimating the EDF of the true, underlying parameters. E!ective histogram estimates result from optimizing a loss function based in a distance between the histogram and its estimate. Similarly, ranking observed data usually produces poor estimates and ranking posterior means can be inappropriate. E!ective estimates should be based on a loss function that caters directly to ranks. Using examples of &borrowing information', shrinkage and the variance/bias trade-o! we motivate Bayes and empirical Bayes analysis. Then, we outline the formal approach and discuss &triple-goal' estimates with values that when ranked produce optimal ranks, for which the EDF is an optimal estimate of the parameter EDF and such that the values themselves are e!ective estimates of co-ordinate-speci"c parameters. We use basic models and data analysis examples to highlight the conceptual and structural issues.
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