Generalized fiducial inference for normal linear mixed models

While linear mixed modeling methods are foundational concepts introduced in any statistical education, adequate general methods for interval estimation involving models with more than a few variance components are lacking, especially in the unbalanced setting. Generalized fiducial inference provides a possible framework that accommodates this absence of methodology. Under the fabric of generalized fiducial inference along with sequential Monte Carlo methods, we present an approach for interval estimation for both balanced and unbalanced Gaussian linear mixed models. We compare the proposed method to classical and Bayesian results in the literature in a simulation study of two-fold nested models and two-factor crossed designs with an interaction term. 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 for small sample sizes. A MATLAB implementation of the proposed algorithm is available from the authors.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1030 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Generalized Fiducial Inference via Discretization

In addition to the usual sources of error that have been long studied by statisticians, many data sets have been rounded off in some manner, either by the measuring device or storage on a computer. In this paper we investigate theoretical properties of generalized fiducial distribution introduced in Hannig (2009) for discretized data. Limit theorems are provided for both fixed sample size with increasing precision of the discretization, and increasing sample size with fixed precision of the discretization. The former provides an attractive definition of generalized fiducial distribution for certain types of exactly observed data overcoming a previous non-uniqueness due to Borel paradox. The latter establishes asymptotic correctness of generalized fiducial inference, in the frequentist, repeated sampling sense, for i.i.d. discretized data under very mild conditions

A fiducial approach to nonparametric deconvolution problem: discrete case

Fiducial inference, as generalized by Hannig et al. (2016), is applied to nonparametric g-modeling (Efron, 2016) in the discrete case. We propose a computationally efficient algorithm to sample from the fiducial distribution, and use generated samples to construct point estimates and confidence intervals. We study the theoretical properties of the fiducial distribution and perform extensive simulations in various scenarios. The proposed approach gives rise to surprisingly good statistical performance in terms of the mean squared error of point estimators and coverage of confidence intervals. Furthermore, we apply the proposed fiducial method to estimate the probability of each satellite site being malignant using gastric adenocarcinoma data with 844 patients (Efron, 2016)

Fiducial inference on the largest mean of a multivariate normal distribution

Inference on the largest mean of a multivariate normal distribution is a surprisingly difficult and unexplored topic. Difficulties arise when two or more of the means are simultaneously the largest mean. Our proposed solution is based on an extension of R.A. Fisher's fiducial inference methods termed generalized fiducial inference. We use a model selection technique along with the generalized fiducial distribution to allow for equal largest means and alleviate the overestimation that commonly occurs. Our proposed confidence intervals for the largest mean have asymptotically correct frequentist coverage and simulation results suggest that they possess promising small sample empirical properties. In addition to the theoretical calculations and simulations we also applied this approach to the air quality index of the four largest cities in the northeastern United States (Baltimore, Boston, New York, and Philadelphia)