Generalized fiducial inference for mixed linear models

Abstract

Fiducial inference was proposed by R.A. Fisher in 1930 to overcome what he perceived as a deficiency in Bayesian methodology -- assuming a prior distribution without prior knowledge. Due to some controversy, fiducial inference quickly fell into disfavor by the statistical community and was left undeveloped by Fisher. There were several attempts over the subsequent decades to revive fiducial inference. Eventually a connection was drawn between fiducial inference and generalized inference, called generalized fiducial inference (GFI). Under the GFI paradigm, inference is performed by considering the generalized fiducial distribution on the parameter space with a flexibility similar to a posterior distribution in the Bayesian framework. GFI can be thought of as a transference of probability from the model space to the parameter space, and a generalized fiducial distribution is defined for the unknown parameters of the model. In this dissertation, we apply the generalized fiducial framework to the normal linear mixed model setting and to logistic models with mixed effects. GFI is a computationally-based mode of inference, and we develop sequential Monte Carlo algorithms to obtain samples from the generalized fiducial distribution on the parameter space. In the normal linear mixed model setting, the proposed method is found to be competitive or better when evaluated based on frequentist criteria of empirical coverage and average length of confidence intervals. In the logistic setting with mixed effects setting, the simulation study reveals that the generalized fiducial approach tends to have correct empirical coverage, along with providing finite confidence intervals in cases where competing methods have disjoint, infinite, or non-calculable intervals. For the final part of this dissertation, we developed a methodology for classifying an unknown powder as a particular harmful substance (Bacillus anthracis spores) or not. A wavelet transformation was incorporated to allow for possible thresholding or standardization, and then a linear model technique using the known elemental structure of the harmful substance was used for dimension reduction, and finally a support vector machine approach was employed for the final classification of the substance. The method was applied to real-data produced from a laser-induced breakdown spectroscopy device.