A Fiducial Approach to Extremes and Multiple Comparisons
Damian Wandler, Ph.D. Candidate, Department of Statistics, Colorado State University
Friday, February 12, 2010
10:00 a.m., Statistics Building, room 008
Generalized fiducial inference is a powerful tool for many difficult problems. Based on an extension of R. A. Fisher's work, we used generalized fiducial inference for two extreme value problems and a multiple comparison procedure.
The first extreme value problem is dealing with the generalized Pareto distribution. The generalized Pareto distribution is relevant to many situations when modeling extremes of random variables. We use a fiducial framework to perform inference on the parameters and the extreme quantiles of the generalized Pareto. This inference technique is demonstrated in both cases when the threshold is a known and unknown parameter. Simmulation results suggest good empirical properties and compared favorably to similar Bayesian and frequentist methods.
The second extreme value problem pertains to the largest mean of a multivariate normal distribution. Difficulties arise when two or more of the means are simultaneously the largest mean. Our solution uses a generalized fiducial distribution and allows for equal largest means to alleviate the overestimation that commonly occurs. Theoretical calculations, simulation results, and application suggest our solution possesses promising asymptotic and empirical properties.
Our solution to the largest mean problem arose from our ability to identify the correct largest mean(s). This essentially became a model selection problem. As a result, we applied a similar model selection approach to the multiple comparison problem. We allowed for all possible groupings (of equality) of the means of k independent normal distributions. Our resulting fiducial probability for the groupings of the means demonstrates the effectiveness of our method by selecting the correct grouping at a high rate.
This dissertation developed solutions to these three problems in detail, however, due to time constraints this talk will focus on the results obtained from our work with the generalized Pareto distribution.
Adviser: Jan Hannig
Co-adviser: Hari Iyer
Committee member: Haonan Wang
Outside committee member: Edwin Chong