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Robust Nonparametric Statistical Methods

Thomas P. Hettmansperger and Joseph W. McKean
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Monographs on Statistics and Applied Probability 119
[Reviewed by
Robert W. Hayden
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There are a variety of textbooks on non-parametric statistics. In the early days, these techniques were defined negatively by their failure to make many assumptions about the population from which a sample was drawn. Then it was realized that many of these techniques amounted to applying parametric techniques to the ranks of the data. Ignoring ties for the moment, the population ranks follow a uniform distribution, in which case we need no assumptions about its shape — we know what it is. The book at hand takes this approach but goes farther and also contrasts parametric and non-parametric statistics in terms of the L2 and L1 norms. That is not a translation into MAA-speak but exactly how the book describes things. So, we might consider this book to be “math. stats. for non-parametrics”. In that category, it hasn’t much competition.

MAA members can be expected to be familiar with the linear algebra and matrix notation in this book, as well as the theorem-proof presentation and the norms involved. A good, modern regression course, with an emphasis on diagnostics and the use of matrices, would be one statistical prerequisite, as the presentation often parallels least squares with alternatives — very helpful for those who already know the least squares side of the story. Familiarity with the common non-parametric techniques would allow one to benefit from the many times they are referenced but not explained in detail. Finally, a mathematical statistics course might contribute knowledge of the sorts of properties statisticians consider desirable in estimators. The theorems proven are not always the ones a mathematician might think worthy of attention, but are driven instead by applications and history.

The main audience for this text is probably statistics departments with Ph.D. programs. MAA members who might be interested include those who are called upon to teach a non-parametric statistics course and who want to understand what is really going on. It carries a significant list of prerequisites, but it does really use and integrate them, making this book a fine capstone course in non-parametric statistics.

After a few years in industry, Robert W. Hayden ( taught mathematics at colleges and universities for 32 years and statistics for 20 years. In 2005 he retired from full-time classroom work. He now teaches statistics online at and does summer workshops for high school teachers of Advanced Placement Statistics. He contributed the chapter on evaluating introductory statistics textbooks to the MAA's Teaching Statistics.

One-Sample Problems
Location Model
Geometry and Inference in the Location Model
Properties of Norm-Based Inference
Robustness Properties of Norm-Based Inference
Inference and the Wilcoxon Signed-Rank Norm
Inference Based on General Signed-Rank Norms
Ranked Set Sampling
L1 Interpolated Confidence Intervals
Two-Sample Analysis

Two-Sample Problems
Geometric Motivation
Inference Based on the Mann-Whitney-Wilcoxon
General Rank Scores
L1 Analyses
Robustness Properties
Proportional Hazards
Two-Sample Rank Set Sampling (RSS)
Two-Sample Scale Problem
Behrens-Fisher Problem
Paired Designs

Linear Models
Geometry of Estimation and Tests
Assumptions for Asymptotic Theory
Theory of Rank-Based Estimates
Theory of Rank-Based Tests
Implementation of the R Analysis
L1 Analysis
Survival Analysis
Correlation Model
High Breakdown (HBR) Estimates
Diagnostics for Differentiating between Fits
Rank-Based Procedures for Nonlinear Models

Experimental Designs: Fixed Effects
One-Way Design
Multiple Comparison Procedures
Two-Way Crossed Factorial
Analysis of Covariance
Further Examples
Rank Transform

Models with Dependent Error Structure
General Mixed Models
Simple Mixed Models
Arnold Transformations
General Estimating Equations (GEE)
Time Series

Multivariate Location Model
Spatial Methods
Affine Equivariant and Invariant Methods
Robustness of Estimates of Location
Linear Model
Experimental Designs

Appendix: Asymptotic Results