##
**Quasi-likelihood and its application. A general approach to optimal parameter estimation.**
*(English)*
Zbl 0879.62076

Springer Series in Statistics. New York, NY: Springer. ix, 235 p. (1997).

The monograph under review is primarily concerned with parameter estimation for a random process \(\{X_t\}\) taking values in \(r\)- dimensional Euclidean space. The distribution of \(X_t\) depends on a characteristic (parameter) \(\theta\) taking values in an open subset of \(p\)-dimensional Euclidean space. The goal is the efficient estimation of \(\theta\) based on a sample \(\{X_t\), \(t\leq T\}\). The author unifies such approaches as least squares and maximum likelihood under the general description of quasi-likelihood. It turns out that the theory needs to be developed in terms of estimating functions rather than the estimators themselves.

The book consists of a Preface and 13 chapters. Chapter 1 is an introduction. In Chapter 2, quasi-likelihood is developed in its general framework. Quasilikelihood estimators are derived from quasi-score estimating functions whose selection involves maximization of a matrix valued information criterion in the partial order of non-negative definite matrices. Both fixed sample and asymptotic formulations are considered. Also treated is the methodology of generalized estimating equations, developed for longitudinal data sets and typically using approximate covariance matrices in the quasi-score estimating function. With the basic formulation provided, the author shows next how a semimartingale model leads to a convenient class of estimating functions of wide applicability. Various illustrations are provided.

Chapter 3 outlines an alternative approach to optimal estimation using estimating functions via the concepts of \(E\)-sufficiency and \(E\)-ancillarity, where \(E\) refers to expectation. Chapter 4 is concerned with asymptotic confidence zones. Under the usual sort of regularity conditions, quasi-likelihood estimators are associated with minimum size asymptotic confidence intervals within their prespecified spaces of estimating functions.

Ordinary quasi-likelihood theory is concerned with the case where the maximum information criterion holds exactly for fixed \(T\) or for each \(T\) as \(T\to\infty\). Chapter 5 deals with the case where optimality holds only in a certain asymptotic sense. This may happen, for instance, when a nuisance parameter is replaced by a consistent estimator. The discussion focuses on situations where the properties of regular quasi-likelihood of consistency and possession of minimum size asymptotic confidence zones are preserved for the estimator. Estimating functions from different sources can conveniently be added, and the issue of their optimal combination is addressed in Chapter 6. Various applications are given.

Chapter 7 deals with projection methods that are useful in situations where a standard application of quasi-likelihood is precluded. Quasi-likelihood approaches are provided for constrained parameter estimation, for estimation in the presence of nuisance parameters, and for generalizing the E-M algorithm for estimation when there are missing data. In Chapter 8 the focus is on deriving the quasi-score estimating function without use of the likelihood, which may be difficult to deal with, or fail to exist. Simple quasi-likelihood derivations of the score functions are provided for estimating the parameters in the covariance matrix, where the distribution is multivariate normal, in diffusion type models, and in hidden Markov random fields.

Chapter 9 deals with issues of hypothesis testing. Generalizations of the classical efficient scores statistic and Wald test statistic are treated. Chapter 10 provides a brief discussion of infinite-dimensional parameter estimation. A sketch is given of the method of sieves, in which the dimension of the parameter is increased as the sample size increases. An informal treatment of estimation in linear semimartingale models, such as occur for counting processes and estimation of the cumulative hazard function, is also provided.

A diverse collection of applications is given in Chapter 11. Estimation is discussed for the mean of a stationary process, a heteroscedastic regression, the infection rate of an epidemic, and a population size via a multiple recapture experiment. Also treated are estimation via robustified estimating functions and recursive estimation. Chapter 12 treats the issues of consistency and asymptotic normality of estimators. The focus here is on martingale based methods, and general forms of martingale strong law and central limit theorems are provided for use in particular cases.

Finally, in Chapter 13 a number of complementary issues involved in the use of quasi-likelihood methods are discussed. The chapter begins with a collection of methods for generating useful families of estimating functions. Integral transform families and the use of the infinitesimal generator of a Markov process are treated. Then, the numerical solution of estimating equations is considered, and methods are examined for dealing with multiple roots when a scalar objective function may not be available. The final section is concerned with resampling methods for the provision of confidence intervals, in particular the jackknife and bootstrap.

The book consists of a Preface and 13 chapters. Chapter 1 is an introduction. In Chapter 2, quasi-likelihood is developed in its general framework. Quasilikelihood estimators are derived from quasi-score estimating functions whose selection involves maximization of a matrix valued information criterion in the partial order of non-negative definite matrices. Both fixed sample and asymptotic formulations are considered. Also treated is the methodology of generalized estimating equations, developed for longitudinal data sets and typically using approximate covariance matrices in the quasi-score estimating function. With the basic formulation provided, the author shows next how a semimartingale model leads to a convenient class of estimating functions of wide applicability. Various illustrations are provided.

Chapter 3 outlines an alternative approach to optimal estimation using estimating functions via the concepts of \(E\)-sufficiency and \(E\)-ancillarity, where \(E\) refers to expectation. Chapter 4 is concerned with asymptotic confidence zones. Under the usual sort of regularity conditions, quasi-likelihood estimators are associated with minimum size asymptotic confidence intervals within their prespecified spaces of estimating functions.

Ordinary quasi-likelihood theory is concerned with the case where the maximum information criterion holds exactly for fixed \(T\) or for each \(T\) as \(T\to\infty\). Chapter 5 deals with the case where optimality holds only in a certain asymptotic sense. This may happen, for instance, when a nuisance parameter is replaced by a consistent estimator. The discussion focuses on situations where the properties of regular quasi-likelihood of consistency and possession of minimum size asymptotic confidence zones are preserved for the estimator. Estimating functions from different sources can conveniently be added, and the issue of their optimal combination is addressed in Chapter 6. Various applications are given.

Chapter 7 deals with projection methods that are useful in situations where a standard application of quasi-likelihood is precluded. Quasi-likelihood approaches are provided for constrained parameter estimation, for estimation in the presence of nuisance parameters, and for generalizing the E-M algorithm for estimation when there are missing data. In Chapter 8 the focus is on deriving the quasi-score estimating function without use of the likelihood, which may be difficult to deal with, or fail to exist. Simple quasi-likelihood derivations of the score functions are provided for estimating the parameters in the covariance matrix, where the distribution is multivariate normal, in diffusion type models, and in hidden Markov random fields.

Chapter 9 deals with issues of hypothesis testing. Generalizations of the classical efficient scores statistic and Wald test statistic are treated. Chapter 10 provides a brief discussion of infinite-dimensional parameter estimation. A sketch is given of the method of sieves, in which the dimension of the parameter is increased as the sample size increases. An informal treatment of estimation in linear semimartingale models, such as occur for counting processes and estimation of the cumulative hazard function, is also provided.

A diverse collection of applications is given in Chapter 11. Estimation is discussed for the mean of a stationary process, a heteroscedastic regression, the infection rate of an epidemic, and a population size via a multiple recapture experiment. Also treated are estimation via robustified estimating functions and recursive estimation. Chapter 12 treats the issues of consistency and asymptotic normality of estimators. The focus here is on martingale based methods, and general forms of martingale strong law and central limit theorems are provided for use in particular cases.

Finally, in Chapter 13 a number of complementary issues involved in the use of quasi-likelihood methods are discussed. The chapter begins with a collection of methods for generating useful families of estimating functions. Integral transform families and the use of the infinitesimal generator of a Markov process are treated. Then, the numerical solution of estimating equations is considered, and methods are examined for dealing with multiple roots when a scalar objective function may not be available. The final section is concerned with resampling methods for the provision of confidence intervals, in particular the jackknife and bootstrap.

Reviewer: J.Melamed (Los Angeles)

### MSC:

62M09 | Non-Markovian processes: estimation |

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

62F12 | Asymptotic properties of parametric estimators |

62M05 | Markov processes: estimation; hidden Markov models |

62F10 | Point estimation |