Statistical Inference The Minimum Distance Approach

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.

IntroductionGeneral Notation Illustrative ExamplesSome Background and Relevant DefinitionsParametric Inference based on the Maximum Likelihood MethodHypothesis Testing by Likelihood MethodsStatistical Functionals and Influence FunctionOutline of the BookStatistical DistancesIntroduction Distances Based on Distribution FunctionsDensity-Based DistancesMinimum Hellinger Distance Estimation: Discrete Models Minimum Distance Estimation Based on Disparities: Discrete ModelsSome Examples Continuous ModelsIntroduction Minimum Hellinger Distance EstimationEstimation of Multivariate Location and Covariance A General StructureThe Basu-Lindsay Approach for Continuous Data Examples Measures of Robustness and Computational IssuesThe Residual Adjustment Function The Graphical Interpretation of RobustnessThe Generalized Hellinger Distance Higher Order Influence AnalysisHigher Order Influence Analysis: Continuous ModelsAsymptotic Breakdown Properties The a-Influence Function Outlier Stability of Minimum Distance Estimators Contamination Envelopes The Iteratively Reweighted Least Squares (IRLS)The Hypothesis Testing ProblemDisparity Difference Test: Hellinger Distance Case Disparity Difference Tests in Discrete Models Disparity Difference Tests: The Continuous Case Power Breakdown of Disparity Difference TestsOutlier Stability of Hypothesis Tests The Two Sample ProblemTechniques for Inlier ModificationMinimum Distance Estimation: Inlier Correction in Small SamplesPenalized Distances Combined Distances o-Combined DistancesCoupled Distances The Inlier-Shrunk Distances Numerical Simulations and ExamplesWeighted Likelihood EstimationThe Discrete Case The Continuous CaseExamples Hypothesis Testing Further ReadingMultinomial Goodness-of-fit TestingIntroduction Asymptotic Distribution of the Goodness-of-Fit StatisticsExact 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 DivergenceThe Minimum L2 Distance EstimatorThe Minimum Density Power Divergence EstimatorA Related Divergence Measure The Censored Survival Data ProblemThe Normal Mixture Model ProblemSelection of Tuning Parameters Other Applications of the Density Power DivergenceOther ApplicationsCensored Data Minimum Hellinger Distance Methods in Mixture ModelsMinimum Distance Estimation Based on Grouped Data Semiparametric Problems Other Miscellaneous Topics Distance Measures in Information and EngineeringIntroduction Entropies and DivergencesCsiszar’s f-DivergenceThe Bregman Divergence Extended f-Divergences Additional Remarks Applications to Other Models Introduction Preliminaries for Other Models Neural NetworksFuzzy Theory Phase RetrievalSummary

Ayanendranath Basu, Hiroyuki Shioya, Chanseok Park