Statistical Inference : The Minimum Distance Approach book cover
1st Edition

Statistical Inference
The Minimum Distance Approach

ISBN 9781420099652
Published June 22, 2011 by Chapman and Hall/CRC
429 Pages 47 B/W Illustrations

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Book Description

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.

Table of Contents

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
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
Minimum Hellinger Distance Estimation
Estimation of Multivariate Location and Covariance
A General Structure
The Basu-Lindsay Approach for Continuous Data

Measures of Robustness and Computational Issues
The Residual Adjustment Function
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
Hypothesis Testing
Further Reading

Multinomial Goodness-of-fit Testing
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
Entropies and Divergences
Csiszar’s f-Divergence
The Bregman Divergence
Extended f-Divergences
Additional Remarks

Applications to Other Models
Preliminaries for Other Models
Neural Networks
Fuzzy Theory
Phase Retrieval

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Ayanendranath Basu, Hiroyuki Shioya, Chanseok Park


"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