zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Nonparametric smoothing and lack-of-fit tests. (English) Zbl 0886.62043
Springer Series in Statistics. New York, NY: Springer. xii, 287 p. DM 79.00; öS 576.70; sFr 72.00; £30.50; $ 44.95 (1997).

The author writes in the preface that the primary aim of the book is to explore the use of nonparametric regression methodology in testing the fit of parametric regression models. Chapters 2 – 4 give a general introduction to estimation of regression curves in the case of single design variables with particular emphasis on smoothing methods. Chapter 2 is an expository introduction to basic methods of nonparametric regression. More attention is paid to kernel methods and Fourier series while splines and local polynomials are discussed only briefly. Chapter 3 studies statistical properties of kernel and Fourier series estimators. Chapter 4 is devoted to the problem of the choice of the estimators’ smoothing parameters and to introduction to several methods of data-driven smoothing.

Chapters 5 – 10 concern the problem of testing the fit of probability models. Chapter 5 reviews classical lack of fit tests, including likelihood ratio tests, reduction methods from linear models and some nonparametric tests. Chapter 6 considers more recently proposed lack of fit tests based on nonparametric linear smoothers. Chapters 7 – 10 are really the most interesting part of the monograph. The lack of fit tests based on data-driven smoothing parameters are studied. Chapters 7 – 8 contain a careful treatment of distributional properties of various “data-driven test” statistics. Chapter 7 introduces tests of “no effects” (regression function is constant). Using the same principles, test procedures for more general types of hypotheses are developed and studied in Chapters 8 and 9. Chapter 9 discusses extensions to multiple regression, testing additivity, testing homoscedasticity and time series trend detection, among others. Chapter 10 provides a number of illustrations of these tests on some data sets.

This is probably the first book dealing with tests of adequacy of parametric function estimates within a nonparametric framework reflecting recent developments in this area. Special attention is paid to tests and estimators based on Fourier coefficients. The book can be read on several levels. Following the advice of the author, the reader can focus only on the applications of estimation and testing procedures (for practitioners) or can learn more theoretical background of the procedures (for more theoretically oriented readers). The book is well written and suitable for graduate students, the material is explained without too many technical details. No excercices or problems to solve are included (this would be useful for graduate students).

MSC:
62G07Density estimation
62G10Nonparametric hypothesis testing
62-01Textbooks (statistics)
62-02Research monographs (statistics)