Date of Award
Doctor of Philosophy
While very useful in the realm of decision theory, it is widely understood that when applied to interval estimation, empirical Bayesian estimation techniques produce intervals with an incorrect width due to the failure to incorporate uncertainty in the estimates of the prior parameters. Traditionally, interval widths have been seen as too short. Various methods have been proposed to address this, with most focusing on the normal model as an application and many attempting to recreate, either naturally or artificially, a hierarchical Bayesian solution. An alternative framework for analysis in the non-normal scenario is proposed and, for the beta-binomial model, it is shown that under this framework the full hierarchical method may produce interval widths that are shorter than empirical Bayesian interval widths. Furthermore, this paper will compare interval widths and frequentist coverage for different Bayesian and non-Bayesian interval correction methods and offer recommendations. This framework may also be extended to the larger natural exponential family with quadratic variance functions, of which the beta-binomial model is a member, and general properties of NEFQVF distributions are given, with a specific application of the gamma-Poisson model. A class of prior is introduced as a limiting state of the framework that, in the hierarchical setting where the shrinkage coefficient is known, extends the well-known conjugacy of NEFQVF families to the hierarchical setting in an approximate way, and intervals are constructed using a refined empirical Bayesian interval correction technique that produce an alternative comparison basis. Coverage and interval widths are shown for this technique for the beta-binomial and gamma-Poisson models. Both produce near-nominal coverage and compare favorably to the full hierarchical solution calculated using MCMC.
As a second topic, a new Bayesian and empirical Bayesian estimate of a baseball team's ``true'' winning percentage is introduced. Common methods for estimating this ``true'' winning percentage, such as the pythagorean expectation or pythagenpat system, rely on the total number of runs scored and allowed over a period of time. A new estimator is proposed that uses independent zero-inflated geometric distributions for runs scored and allowed per inning to determine a winning percentage. This estimator outperforms methods based on total runs scored and allowed in terms of mean-squared error using actual win totals. Interval estimation for this estimator is directly shown using frequentist or Bayesian techniques. Empirical Bayesian techniques are shown as an approximation to the full hierarchical solution. Selected interval widths are compared using all three methods, with slight differences in the shrinkage amount given a full season's worth of data.
Robert Christian Foster
Foster, Robert Christian, "Topics in empirical Bayesian analysis" (2016). Graduate Theses and Dissertations. 15910.