Heyde Christopher С. Quasi-Likelihood And Its Application - A General Approach To Optimal Parameter Estimation

Springer, 1997. — 236 p. — (Springer Series in Statistics). — ISBN13: 978-0387982250; ISBN10: 0387982256.This book is concerned with the general theory of optimal estimation of parameters in systems subject to random effects and with the application of this theory. The focus is on choice of families of estimating functions, rather than the estimators derived therefrom, and on optimization within these families. Only assumptions about means and covariances are required for an initial discussion. Nevertheless, the theory that is developed mimics that of maximum likelihood, at least to the first order of asymptotics. The term quasi-likelihood has often had a narrow interpretation, associated with its application to generalized linear model type contexts, while that of optimal estimating functions has embraced a broader concept. There is, however, no essential distinction between the underlying ideas and the term quasi-likelihood has herein been adopted as the general label. This emphasizes its role in extension of likelihood based theory. The idea throughout involves finding quasi-scores from families of estimating functions. Then, the quasilikelihood estimator is derived from the quasi-score by equating to zero and solving, just as the maximum likelihood estimator is derived from the likelihood score.The emphasis in the monograph is on concepts rather than on mathematical theory. Indeed, formalities have been suppressed to avoid obscuring “typical” results with the phalanx of regularity conditions and qualifiers necessary to avoid the usual uninformative types of counterexamples which detract from most statistical paradigms. In discussing theory which holds to the first order of asymptotics the treatment is especially informal, as befits the context. Sufficient conditions which ensure the behaviour described are not difficult to furnish but are fundamentally uninlightening.Many examples of the use of the methods in both classical statistical and stochastic process contexts are provided. A collection of complements and exercises has been included to make the material more useful in a teaching environment and the book should be suitable for advanced courses and seminars. Prerequisites are sound basic courses in measure theoretic probability and in statistical inference.