Nonlinear statistical models.

*(English)*Zbl 0611.62071
Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. New York etc.: John Wiley & Sons. XII, 610 p. £48.00 (1987).

During the last several years, linear modeling has played a major role in statistical data analysis due to its easy accessibility for statistical computing purposes. In view of availability of computers and their capability for intricate computations at high speed, nonlinear modeling has found use for statistical modeling. There has been a large amount of theoretical work in deriving properties of various type of estimators in nonlinear regression. Here the author brings together the recent advances in this important area. His main interest is in applications to econometrics. The list of contents is as follows. 1. Univariate nonlinear regression, 2. Univariate nonlinear regression: Special situations, 3. A unified asymptotic theory of nonlinear models with regression structure, 4. Univariate nonlinear regression: asymptotic theory, 5. Multivariate nonlinear regression, 6. Nonlinear simultaneous equations models, 7. A unified asymptotic theory for dynamic nonlinear models.

In Chapter 1, the classical results known for linear models are extended to nonlinear models when the errors are independent and identically distributed. Chapter 2 contains discussion on estimates of parameters when the errors are heteroscedastic or correlated and on testing nonlinear specifications. Asymptotic theory for nonlinear regression estimators is studied in Chapters 3 and 4. Two types of estimators are studied: (i) Least mean distance estimators, (ii) Method of moments estimators. Since the author assumes strong regularity conditions involving differentiability of the regression function, one should caution that least mean distance estimator is not the same as least absolute deviation estimator discussed in the literature. It is surprising that no mention is made of the recent work of C.-F. Wu [Ann. Stat. 9, 501-513 (1981; Zbl 0475.62050)] on asymptotic theory of nonlinear least squares estimation. An alternate approach to study asymptotic theory of estimation for nonlinear regression models is given in Chapter 7 of the reviewer’s book ”Asymptotic theory of statistical inference”. (1987; Zbl 0604.62025). Chapter 5 contains extensions of results in Chapter 1 to multivariate nonlinear regression. Results on nonlinear simultaneous equation models useful for econometricians are given in Chapter 6. Chapter 7 contains asymptotic theory for dynamic nonlinear models.

The book has a large number of worked out examples along with computations using SAS. Each chapter contains a problem section at the end and the problems contain hints useful in completing the gaps in the proofs given in the main section. Algorithms for programming are provided at several places. Due to the nature of notation involved, the proofs at times become lengthy and clumsy. It should be possible to simplify the proofs and the notation and computations at several places using algebraic methods via matrix theory and statistical results via Cramer- Wold technique in multivariate problems for instance. On pages 155-157, the author considers an infinite dimensional parameter space \(\Gamma\) to discuss data generating models. He says that ”no generality is lost by assuming that \(\Gamma\) is compact”. This is not correct. For instance, if \(\Gamma\) is a Hilbert space and if B is a locally compact subspace of \(\Gamma\), then B has to be finite dimensional!

It seems that the book is written with two aims in mind: first to provide the latest techniques and results in the subject useful for econometricians for nonlinear modeling and second to give the extensive reasoning and the proofs of these results. The author seems to have succeeded in the former but not in the latter. A reader interested in technical details with more illuminating proofs and reasons should look elsewhere for a more comprehensive and interesting treatment. The book is not a text book but it is clearly a useful addition to the recent literature on nonlinear modeling.

In Chapter 1, the classical results known for linear models are extended to nonlinear models when the errors are independent and identically distributed. Chapter 2 contains discussion on estimates of parameters when the errors are heteroscedastic or correlated and on testing nonlinear specifications. Asymptotic theory for nonlinear regression estimators is studied in Chapters 3 and 4. Two types of estimators are studied: (i) Least mean distance estimators, (ii) Method of moments estimators. Since the author assumes strong regularity conditions involving differentiability of the regression function, one should caution that least mean distance estimator is not the same as least absolute deviation estimator discussed in the literature. It is surprising that no mention is made of the recent work of C.-F. Wu [Ann. Stat. 9, 501-513 (1981; Zbl 0475.62050)] on asymptotic theory of nonlinear least squares estimation. An alternate approach to study asymptotic theory of estimation for nonlinear regression models is given in Chapter 7 of the reviewer’s book ”Asymptotic theory of statistical inference”. (1987; Zbl 0604.62025). Chapter 5 contains extensions of results in Chapter 1 to multivariate nonlinear regression. Results on nonlinear simultaneous equation models useful for econometricians are given in Chapter 6. Chapter 7 contains asymptotic theory for dynamic nonlinear models.

The book has a large number of worked out examples along with computations using SAS. Each chapter contains a problem section at the end and the problems contain hints useful in completing the gaps in the proofs given in the main section. Algorithms for programming are provided at several places. Due to the nature of notation involved, the proofs at times become lengthy and clumsy. It should be possible to simplify the proofs and the notation and computations at several places using algebraic methods via matrix theory and statistical results via Cramer- Wold technique in multivariate problems for instance. On pages 155-157, the author considers an infinite dimensional parameter space \(\Gamma\) to discuss data generating models. He says that ”no generality is lost by assuming that \(\Gamma\) is compact”. This is not correct. For instance, if \(\Gamma\) is a Hilbert space and if B is a locally compact subspace of \(\Gamma\), then B has to be finite dimensional!

It seems that the book is written with two aims in mind: first to provide the latest techniques and results in the subject useful for econometricians for nonlinear modeling and second to give the extensive reasoning and the proofs of these results. The author seems to have succeeded in the former but not in the latter. A reader interested in technical details with more illuminating proofs and reasons should look elsewhere for a more comprehensive and interesting treatment. The book is not a text book but it is clearly a useful addition to the recent literature on nonlinear modeling.

Reviewer: B.L.S.Prakasa Rao

##### MSC:

62J02 | General nonlinear regression |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62P20 | Applications of statistics to economics |