In the book A Parametric Approach to Nonparametric Statistics, the authors compile the research results and approaches from literature that combine parametric techniques and nonparametric statistics to bridge the gap between these two branches of statistics. Nonparametric statistics being a large research area, and of course, parametric statistics being the classical and well established subject, such a work is clearly unique and relevant. The book is divided into three parts, roughly covering background, theory, and applications, respectively.
First part of this book is a review of fundamental concepts of probability and statistics like parameter estimation, hypothesis testing, likelihood, Bayesian methods, and variational inference. There is also a brief refresher on some nonparametric statistical ideas like linear rank statistics, U statistics, and Hoeffding’s combinatorial central limit theorem.
The second part of this book deals with the core theory of nonparametric statistics. This part starts with goodness-of-fit tests with some discussion on smooth tests. The authors discuss motivation and theory behind Neyman’s smooth tests. Smooth tests for other type of data are also considered, for example, models for discrete distributions, categorical data and ranking data are discussed in detail.
The authors then move on to one-sample and two-sample testing problems. They demonstrate in detail similar approaches, using parametric techniques like composite likelihood and deriving a limiting distribution using nonparametric tools like U statistics or rank statistics. The test setups considered are: sign test and Wilcoxon signed rank test among one-sample tests, and for two-sample problems: permutation tests, Mann–Whitney–Wilcoxon rank sum test. Tests for equality of scale parameter are also considered.
The next chapter presents a unified theory of multiple sample hypothesis testing based on ranks. Two sets of ranks are constructed based on alternative and the data, with the test statistic defined by a measure of distance between the sets. The authors considered multisample problem in ordered and unordered case, also tests against umbrella alternatives, dispersion. There is also discussion of equality of several independent samples and tests for interactions.
In the following chapter, the authors consider additional applications of the smooth model paradigm. Tests for trend are the first application they at, followed by tests for concordance with application using Spearman scores, Kendall scores and Hamming scores. More general tests related to design problems are discussed including generalized block designs.
The final two chapters of this part deal with optimal rank tests and concepts of efficiency. Chapter 8 shows that locally most powerful tests as an application of the general unified theory of multiple sample tests. It also considers regression problem and optimal choice of score function for complete and incomplete block design problems. The final chapter provides discussion on asymptotic efficiency of various tests considered in previous chapters.
The third part of this book looks at modern research applications of nonparametric statistics. In the opinion of this reviewer, this part is the most interesting part of the book as it deals with new research problems and applications with data analysis and simulation details.
First, the authors consider the problem of multiple change point detection as a candidate for parametric formulation with smooth alternative. They discuss the construction of the composite likelihood, estimation of the change points, testing significance and theoretical consistency including some simulation experiments and real data analysis.
In Chapter 11, the authors discuss their work on angle-based models for ranking data. They perform a Bayesian analysis including conjugate prior and variational inference. They also provide real data applications with two datasets.
The final chapter deals with analysis of censored data and demonstrates that the parametric embedding is applicable in this domain as well. After a brief review on survival analysis, the authors provide some guidelines on the appropriate choice of the parametric family and finally concluding with several important research works on parametric embedding for censored and truncated data.
Overall this book provides an excellent summary of the developments of applications of parametric methods on problems of nonparametric statistics. The book can serve as a literature guide for researchers in both nonparametric and parametric statistics as well as can be used as a reference book on any graduate course on nonparametric statistics. The authors comment that this book is meant to be a bridge between these two branches of statistics and combining the best aspects of both. This is very clearly achieved.
Bosch Center for Artificial Intelligence