Jan Hannig

Fiducial Generalized Confidence Intervals

Generalized pivotal quantities (GPQs) and generalized confidence intervals (GCIs) have proven to be useful tools for making inferences in many practical problems. Although GCIs are not guaranteed to have exact frequentist coverage, a number of published and unpublished simulation studies suggest that the coverage probabilities of such intervals are sufficiently close to their nominal value so as to be useful in practice. In this article we single out a subclass of generalized pivotal quantities, which we call fiducial generalized pivotal quantities (FGPQs), and show that under some mild conditions, GCIs constructed using FGPQs have correct frequentist coverage, at least asymptotically. We describe three general approaches for constructing FGPQs—a recipe based on invertible pivotal relationships, and two extensions of it—and demonstrate their usefulness by deriving some previously unknown GPQs and GCIs. It is fair to say that nearly every published GCI can be obtained using one of these recipes. As an interesting byproduct of our investigations, we note that the subfamily of fiducial generalized pivots has a close connection with fiducial inference proposed by R. A. Fisher. This is why we refer to the proposed generalized pivots as fiducial generalized pivotal quantities. We demonstrate these concepts using several examples.

Advanced Distribution Theory for SiZer

SiZer is a powerful method for exploratory data analysis. In this article approximations to the distributions underlying the simultaneous statistical inference are investigated, and large improvements are made in the approximation using extreme value theory. This results in improved size, and also in an improved global inference version of SiZer. The main points are illustrated with real data and simulated examples.

Fiducial Intervals for Variance Components in an Unbalanced Two-Component Normal Mixed Linear Model

In this article we propose a new method for constructing confidence intervals for σα2,σϵ2, and the intraclass correlation ρ==σα2(σα2++σε2) in a two-component mixed-effects linear model. This method is based on an extension of R. A. Fishers fiducial argument. We conducted a simulation study to compare the resulting interval estimates with other competing confidence interval procedures from the literature. Our results demonstrate that the proposed fiducial intervals have satisfactory performance in terms of coverage probability, as well as shorter average confidence interval lengths overall. We also prove that these fiducial intervals have asymptotically exact frequentist coverage probability. The computations for the proposed procedures are illustrated using real data examples.

Support vector machine classification of suspect powders using laser-induced breakdown spectroscopy (LIBS) spectral data

Classification of suspect powders, by using laser‐induced breakdown spectroscopy (LIBS) spectra, to determine if they could contain Bacillus anthracis spores is difficult because of the variability in their composition and the variability typically associated with LIBS analysis. A method that builds a support vector machine classification model for such spectra relying on the known elemental composition of the Bacillus spores was developed. A wavelet transformation was incorporated in this method to allow for possible thresholding or standardization, then a linear model technique using the known elemental structure of the spores was incorporated for dimension reduction, and a support vector machine approach was employed for the final classification of the substance. The method was applied to real data produced from an LIBS device. Several methods used to test the predictive performance of the classification model revealed promising results. Published 2012. This article is a US Government work and is in the public domain in the USA.

Robust SiZer for Exploration of Regression Structures and Outlier Detection

The SiZer methodology is a valuable tool for conducting exploratory data analysis. In this article a robust version of SiZer is developed for the regression setting. This robust SiZer is capable of producing SiZer maps with different degrees of robustness. By inspecting such SiZer maps, either as a series of plots or in the form of a movie, the structures hidden in a dataset can be more effectively revealed. It is also demonstrated that the robust SiZer can be used to help identify outliers. Results from both real data and simulated examples will be provided.

Generalized Fiducial Inference: A Review and New Results

Abstract R. A. Fisher, the father of modern statistics, proposed the idea of fiducial inference during the first half of the 20th century. While his proposal led to interesting methods for quantifying uncertainty, other prominent statisticians of the time did not accept Fisher’s approach as it became apparent that some of Fisher’s bold claims about the properties of fiducial distribution did not hold up for multi-parameter problems. Beginning around the year 2000, the authors and collaborators started to reinvestigate the idea of fiducial inference and discovered that Fisher’s approach, when properly generalized, would open doors to solve many important and difficult inference problems. They termed their generalization of Fisher’s idea as generalized fiducial inference (GFI). The main idea of GFI is to carefully transfer randomness from the data to the parameter space using an inverse of a data-generating equation without the use of Bayes’ theorem. The resulting generalized fiducial distribution (GFD) can then be used for inference. After more than a decade of investigations, the authors and collaborators have developed a unifying theory for GFI, and provided GFI solutions to many challenging practical problems in different fields of science and industry. Overall, they have demonstrated that GFI is a valid, useful, and promising approach for conducting statistical inference. The goal of this article is to deliver a timely and concise introduction to GFI, to present some of the latest results, as well as to list some related open research problems. It is authors’ hope that their contributions to GFI will stimulate the growth and usage of this exciting approach for statistical inference. Supplementary materials for this article are available online.

Fiducial Approach to Uncertainty Assessment Accounting for Error due to Instrument Resolution

This paper presents an approach for making inference on the parameters µ and σ of a Gaussian distribution in the presence of resolution errors. The approach is based on the principle of fiducial inference and requires a Monte Carlo method for computing uncertainty intervals. A small simulation study is carried out to evaluate the performance of the proposed procedure and compare it with some of the existing procedures. The results indicate that the fiducial procedure is comparable to the best of the competing procedures for inference on µ. However, unlike some of the competing procedures, the same Monte Carlo calculations also provide inference for σ and many other related quantities of interest. (Some figures in this article are in colour only in the electronic version)

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.

A note on Dempster-Shafer recombination of confidence distributions

It is often the case that there are several studies measuring the same parameter. Naturally, it is of interest to provide a systematic way to combine the information from these studies. Examples of such situa- tions include clinical trials, key comparison trials and other problems of practical importance. Singh et al. (2005) provide a compelling framework for combining information from multiple sources using the framework of confidence distributions. In this paper we investigate the feasibility of us- ing the Dempster-Shafer recombination rule on this problem. We derive a practical combination rule and show that under assumption of asymptotic normality, the Dempster-Shafer combined confidence distribution is asymp- totically equivalent to one of the method proposed in Singh et al. (2005). Numerical studies and comparisons for the common mean problem and the odds ratio in 2 × 2 tables are included.

Multiscale Exploratory Analysis of Regression Quantiles Using Quantile SiZer

The SiZer methodology proposed by Chaudhuri and Marron (1999) is a valuable tool for conducting exploratory data analysis. Since its inception different versions of SiZer have been proposed in the literature. Most of these SiZer variants are targeting the mean structure of the data, and are incapable of providing any information about the quantile composition of the data. To fill this need, this article proposes a quantile version of SiZer for the regression setting. By inspecting the SiZer maps produced by this new SiZer, real quantile structures hidden in a dataset can be more effectively revealed, while at the same time spurious features can be filtered out. The utility of this quantile SiZer is illustrated via applications to both real data and simulated examples. This article has supplementary material online.