An Introduction to Nonparametric Statistics (Chapman & Hall/CRC Texts in Statistical Science)

✍ Scribed by John E. Kolassa

• Publisher: Chapman and Hall/CRC
• Year: 2020
• Tongue: English
• Leaves: 225
• Edition: 1
• Category: Library

✦ Synopsis

An Introduction to Nonparametric Statistics presents techniques for statistical analysis in the absence of strong assumptions about the distributions generating the data. Rank-based and resampling techniques are heavily represented, but robust techniques are considered as well. These techniques include one-sample testing and estimation, multi-sample testing and estimation, and regression.

Attention is paid to the intellectual development of the field, with a thorough review of bibliographical references. Computational tools, in R and SAS, are developed and illustrated via examples. Exercises designed to reinforce examples are included.

Features

• Rank-based techniques including sign, Kruskal-Wallis, Friedman, Mann-Whitney and Wilcoxon tests are presented

• Tests are inverted to produce estimates and confidence intervals

• Multivariate tests are explored

• Techniques reflecting the dependence of a response variable on explanatory variables are presented

• Density estimation is explored

• The bootstrap and jackknife are discussed

This text is intended for a graduate student in applied statistics. The course is best taken after an introductory course in statistical methodology, elementary probability, and regression. Mathematical prerequisites include calculus through multivariate differentiation and integration, and, ideally, a course in matrix algebra.

✦ Table of Contents

Cover
Half Title
Series Page
Title Page
Copyright Page
Contents
Introduction
1. Background
1.1 Probability Background
1.1.1 Probability Distributions for Observations
1.1.1.1 Gaussian Distribution
1.1.1.2 Uniform Distribution
1.1.1.3 Laplace Distribution
1.1.1.4 Cauchy Distribution
1.1.1.5 Logistic Distribution
1.1.1.6 Exponential Distribution
1.1.2 Location and Scale Families
1.1.3 Sampling Distributions
1.1.3.1 Binomial Distribution
1.1.4 ˜2-distribution
1.1.5 T-distribution
1.1.6 F-distribution
1.2 Elementary Tasks in Frequentist Inference
1.2.1 Hypothesis Testing
1.2.1.1 One-Sided Hypothesis Tests
1.2.1.2 Two-Sided Hypothesis Tests
1.2.1.3 P-values
1.2.2 Confidence Intervals
1.2.2.1 P-value Inversion
1.2.2.2 Test Inversion with Pivotal Statistics
1.2.2.3 A Problematic Example
1.3 Exercises
2. One-Sample Nonparametric Inference
2.1 Parametric Inference on Means
2.1.1 Estimation Using Averages
2.1.2 One-Sample Testing for Gaussian Observations
2.2 The Need for Distribution-Free Tests
2.3 One-Sample Median Methods
2.3.1 Estimates of the Population Median
2.3.2 Hypothesis Tests Concerning the Population Median
2.3.3 Con dence Intervals for the Median
2.3.4 Inference for Other Quantiles
2.4 Comparing Tests
2.4.1 Power, Sample Size, and Effect Size
2.4.1.1 Power
2.4.1.2 Sample and Effect Sizes
2.4.2 Effciency Calculations
2.4.3 Examples of Power Calculations
2.5 Distribution Function Estimation
2.6 Exercises
3. Two-Sample Testing
3.1 Two-Sample Approximately Gaussian Inference
3.1.1 Two-Sample Approximately Gaussian Inference on Expectations
3.1.2 Approximately Gaussian Dispersion Inference
3.2 General Two-Sample Rank Tests
3.2.1 Null Distributions of General Rank Statistics
3.2.2 Moments of Rank Statistics
3.3 A First Distribution-Free Test
3.4 The Mann-Whitney-Wilcoxon Test
3.4.1 Exact and Approximate Mann-Whitney Probabilities
3.4.1.1 Moments and Approximate Normality
3.4.2 Other Scoring Schemes
3.4.3 Using Data as Scores: the Permutation Test
3.5 Empirical Levels and Powers of Two-Sample Tests
3.6 Adaptation to the Presence of Tied Observations
3.7 Mann-Whitney-Wilcoxon Null Hypotheses
3.8 E ciency and Power of Two-Sample Tests
3.8.1 Efficacy of the Gaussian-Theory Test
3.8.2 Efficacy of the Mann-Whitney-Wilcoxon Test
3.8.3 Summarizing Asymptotic Relative Efficiency
3.8.4 Power for Mann-Whitney-Wilcoxon Testing
3.9 Testing Equality of Dispersion
3.10 Two-Sample Estimation and Con dence Intervals
3.10.1 Inversion of the Mann-Whitney-Wilcoxon Test
3.11 Tests for Broad Alternatives
3.12 Exercises
4. Methods for Three or More Groups
4.1 Gaussian-Theory Methods
4.1.1 Contrasts
4.1.2 Multiple Comparisons
4.2 General Rank Tests
4.2.1 Moments of General Rank Sums
4.2.2 Construction of a Chi-Square-Distributed Statistic
4.3 The Kruskal-Wallis Test
4.3.1 Kruskal-Wallis Approximate Critical Values
4.4 Other Scores for Multi-Sample Rank Based Tests
4.5 Multiple Comparisons
4.6 Ordered Alternatives
4.7 Powers of Tests
4.7.1 Power of Tests for Ordered Alternatives
4.7.2 Power of Tests for Unordered Alternatives
4.8 E ciency Calculations
4.8.1 Ordered Alternatives
4.8.2 Unordered Alternatives
4.9 Exercises
5. Group Differences with Blocking
5.1 Gaussian Theory Approaches
5.1.1 Paired Comparisons
5.1.2 Multiple Group Comparisons
5.2 Nonparametric Paired Comparisons
5.2.1 Estimating the Population Median Di erence
5.2.2 Con dence Intervals
5.2.3 Signed-Rank Statistic Alternative Distribution
5.3 Two-Way Non-Parametric Analysis of Variance
5.3.1 Distribution of Rank Sums
5.4 A Generalization of the Test of Friedman
5.4.1 The Balanced Case
5.4.2 The Unbalanced Case
5.5 Multiple Comparisons and Scoring
5.6 Tests for a Putative Ordering in Two-Way Layouts
5.7 Exercises
6. Bivariate Methods
6.1 Parametric Approach
6.2 Permutation Inference
6.3 Nonparametric Correlation
6.3.1 Rank Correlation
6.3.1.1 Alternative Expectation of the Spearman Correlation
6.3.2 Kendall's
6.4 Bivariate Semi-Parametric Estimation via Correlation
6.4.1 Inversion of the Test of Zero Correlation
6.4.1.1 Inversion of the Pearson Correlation
6.4.1.2 Inversion of Kendall's T
6.4.1.3 Inversion of the Spearman Correlation
6.5 Exercises
7. Multivariate Analysis
7.1 Standard Parametric Approaches
7.1.1 Multivariate Estimation
7.1.2 One-Sample Testing
7.1.3 Two-Sample Testing
7.2 Nonparametric Multivariate Estimation
7.2.1 Equivariance Properties
7.3 Nonparametric One-Sample Testing Approaches
7.3.1 More General Permutation Solutions
7.4 Con dence Regions for a Vector Shift Parameter
7.5 Two-Sample Methods
7.5.1 Hypothesis Testing
7.5.1.1 Permutation Testing
7.5.1.2 Permutation Distribution Approximations
7.6 Exercises
8. Density Estimation
8.1 Histograms
8.2 Kernel Density Estimates
8.3 Exercises
9. Regression Function Estimates
9.1 Standard Regression Inference
9.2 Kernel and Local Regression Smoothing
9.3 Isotonic Regression
9.4 Splines
9.5 Quantile Regression
9.5.1 Fitting the Quantile Regression Model
9.6 Resistant Regression
9.7 Exercises
10. Resampling Techniques
10.1 The Bootstrap Idea
10.1.1 The Bootstrap Sampling Scheme
10.2 Univariate Bootstrap Techniques
10.2.1 The Normal Method
10.2.2 Basic Interval
10.2.3 The Percentile Method
10.2.4 BCa Method
10.2.5 Summary So Far, and More Examples
10.3 Bootstrapping Multivariate Data Sets
10.3.1 Regression Models and the Studentized Bootstrap Method
10.3.2 Fixed
10.4 The Jackknife
10.4.1 Examples of Biases of the Proper Order
10.4.2 Bias Correction
10.4.2.1 Correcting the Bias in Mean Estimators
10.4.2.2 Correcting the Bias in Quantile Estimators
10.5 Exercises
Appendix A: Analysis Using the SAS System
Appendix B: Construction of Heuristic Tables and Figures Using R
Bibliography
Index