# Statistical Inference

## The Minimum Distance Approach

© 2011 – Chapman and Hall/CRC

429 pages | 47 B/W Illus.

Hardback: 9781420099652
pub: 2011-06-21
US Dollars\$104.95
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In many ways, estimation by an appropriate minimum distance method is one of the most natural ideas in statistics. However, there are many different ways of constructing an appropriate distance between the data and the model: the scope of study referred to by "Minimum Distance Estimation" is literally huge. Filling a statistical resource gap, Statistical Inference: The Minimum Distance Approach comprehensively overviews developments in density-based minimum distance inference for independently and identically distributed data. Extensions to other more complex models are also discussed.

Comprehensively covering the basics and applications of minimum distance inference, this book introduces and discusses:

• The estimation and hypothesis testing problems for both discrete and continuous models
• The robustness properties and the structural geometry of the minimum distance methods
• The inlier problem and its possible solutions, and the weighted likelihood estimation problem
• The extension of the minimum distance methodology in interdisciplinary areas, such as neural networks and fuzzy sets, as well as specialized models and problems, including semi-parametric problems, mixture models, grouped data problems, and survival analysis.

Statistical Inference: The Minimum Distance Approach gives a thorough account of density-based minimum distance methods and their use in statistical inference. It covers statistical distances, density-based minimum distance methods, discrete and continuous models, asymptotic distributions, robustness, computational issues, residual adjustment functions, graphical descriptions of robustness, penalized and combined distances, weighted likelihood, and multinomial goodness-of-fit tests. This carefully crafted resource is useful to researchers and scientists within and outside the statistics arena.

### Reviews

"The book is an excellent and thorough outline of work in the area. It would provide an ideal volume for someone who plans to undertake research in the area."

International Statistical Review, 2013

"The book provides a comprehensive overview of the theory of density-based minimum distance methods and it is well written and easy to read and understand. The book is well suited for graduate students, professionals and researchers not only in statistics but also in biosciences, engineering and various other fields where statistical inference plays a fundamental role."

—Alex Karagrigoriou, Journal of Applied Statistics, 2012

Introduction

General Notation

Illustrative Examples

Some Background and Relevant Definitions

Parametric Inference based on the Maximum Likelihood Method

Hypothesis Testing by Likelihood Methods

Statistical Functionals and Influence Function

Outline of the Book

Statistical Distances

Introduction

Distances Based on Distribution Functions

Density-Based Distances

Minimum Hellinger Distance Estimation: Discrete Models

Minimum Distance Estimation Based on Disparities: Discrete Models

Some Examples

Continuous Models

Introduction

Minimum Hellinger Distance Estimation

Estimation of Multivariate Location and Covariance

A General Structure

The Basu-Lindsay Approach for Continuous Data

Examples

Measures of Robustness and Computational Issues

The Graphical Interpretation of Robustness

The Generalized Hellinger Distance

Higher Order Influence Analysis

Higher Order Influence Analysis: Continuous Models

Asymptotic Breakdown Properties

The α-Influence Function

Outlier Stability of Minimum Distance Estimators

Contamination Envelopes

The Iteratively Reweighted Least Squares (IRLS)

The Hypothesis Testing Problem

Disparity Difference Test: Hellinger Distance Case

Disparity Difference Tests in Discrete Models

Disparity Difference Tests: The Continuous Case

Power Breakdown of Disparity Difference Tests

Outlier Stability of Hypothesis Tests

The Two Sample Problem

Techniques for Inlier Modification

Minimum Distance Estimation: Inlier Correction in Small Samples

Penalized Distances

Combined Distances

ǫ-Combined Distances

Coupled Distances

The Inlier-Shrunk Distances

Numerical Simulations and Examples

Weighted Likelihood Estimation

The Discrete Case

The Continuous Case

Examples

Hypothesis Testing

Multinomial Goodness-of-fit Testing

Introduction

Asymptotic Distribution of the Goodness-of-Fit Statistics

Exact Power Comparisons in Small Samples

Choosing a Disparity to Minimize the Correction Terms

Small Sample Comparisons of the Test Statistics

Inlier Modified Statistics

An Application: Kappa Statistics

The Density Power Divergence

The Minimum L2 Distance Estimator

The Minimum Density Power Divergence Estimator

A Related Divergence Measure

The Censored Survival Data Problem

The Normal Mixture Model Problem

Selection of Tuning Parameters

Other Applications of the Density Power Divergence

Other Applications

Censored Data

Minimum Hellinger Distance Methods in Mixture Models

Minimum Distance Estimation Based on Grouped Data

Semiparametric Problems

Other Miscellaneous Topics

Distance Measures in Information and Engineering

Introduction

Entropies and Divergences

Csiszar’s f-Divergence

The Bregman Divergence

Extended f-Divergences

Applications to Other Models

Introduction

Preliminaries for Other Models

Neural Networks

Fuzzy Theory

Phase Retrieval

Summary