Thas Olivier. Comparing Distributions

Springer, 2009. — 353 p. — (Springer Series in Statistics). — ISBN 978-0-387-92709-1.Comparing Distributions refers to the statistical data analysis that encompasses the traditional goodness-of-fit testing. Whereas the latter includes only formal statistical hypothesis tests for the one-sample and the K-sample problems, this book presents a more general and informative treatment by also considering graphical and estimation methods.One-Sample ProblemsThe History of the One-Sample GOF Problem
Example Datasets
The Pearson Chi-Squared Test
Pearson X2 Tests for Continuous Distributions
Preliminaries (Building Blocks)
The Empirical Distribution Function
Empirical Processes
The Quantile Function and the Quantile Process
Comparison Distribution
Hilbert Spaces
Orthonormal Functions
Parameter Estimation
Nonparametric Density Estimation
Hypothesis Testing
Graphical Tools
Histograms and Box Plots
Probability Plots and Comparison Distribution
Comparison Distribution
Smooth Tests
Smooth Models
Smooth Tests
Adaptive Smooth Tests
Smooth Tests for Discrete Distributions
A Semiparametric Framework
Example
Some Practical Guidelines for Smooth Tests
Methods Based on the Empirical Distribution Function
The Kolmogorov–Smirnov Test
Tests as Integrals of Empirical Processes
Generalisations of EDF Tests
The Sample Space Partition Tests
Some Further Bibliographic Notes
Some Practical Guidelines for EDF Tests
Two-Sample and K-Sample Problems
Introduction
The Problem Defined
Example Datasets
Preliminaries (Building Blocks)
Permutation Tests
Linear Rank Tests
The Pooled Empirical Distribution Function
The Comparison Distribution
The Quantile Process
Stochastic Ordering and Related Properties
Graphical Tools
PP and QQ Plots
Comparisons Distributions
Some Important Two-Sample Tests
The Relation Between Statistical Tests and Hypotheses
The Wilcoxon Rank Sum and the Mann–Whitney Tests
The Diagnostic Property of Two-Sample Tests
Optimal Linear Rank Tests for Normal Location-Shift Models
Rank Tests for Scale Differences
The Kruskal–Wallis Test and the ANOVA F-Test
Some Final Remarks
Smooth Tests
Smooth Tests for the 2-Sample Problem
The Diagnostic Property
Smooth Tests for the K-Sample Problem
Adaptive Smooth Tests
Examples
Smooth Tests That Are Not Based on Ranks
Some Practical Guidelines for Smooth Tests
Methods Based on the Empirical Distribution Function
The Two-Sample and K-Sample Kolmogorov–Smirnov Test
Tests of the Anderson–Darling Type
Adaptive Tests of Neuhaus
Some Practical Guidelines for EDF Tests
Two Final Methods and Some Final Thoughts
A Contigency Table Approach
The Sample Space Partition Tests
Some Final Thoughts and Conclusions
Proofs
Proof of Theorem 1.1
Proof of Theorem 1.2
Proof of Theorem 4.1
Proof of Lemma 4.1
Proof of Lemma 4.2
Proof of Lemma 4.3
Proof of Theorem 4.10
Proof of Theorem 4.2
Heuristic Proof of Theorem 5.2
Proof of Theorem 9.1
The Bootstrap and Other Simulation Techniques
Simulation of EDF Statistics Under the Simple Null Hypothesis
The Parametric Bootstrap for Composite Null Hypotheses
A Modified Nonparametric Bootstrap for Testing Semiparametric Null Hypotheses