“Our aim in this book is to study the asymptotic theory of statistical inference for semimartingale models in a unified manner.” (From the author’s preface.) The text is organized in 10 chapters with separate lists of references. The following themes are treated (the numbers in the parentheses give the number of pages/number of references):
1. Semimartingales (100/70); 2. Exponential families of stochastic processes (20/8); 3. Asymptotic likelihood theory (30/20); 4. Asymptotic likelihood theory for diffusions with jumps (38/25); 5. Quasi likelihood and semimartingales (32/24); 6. Local asymptotic behavior of semimartingale experiments (58/23); 7. Likelihood and asymptotic efficiency (52/16); 8. Inference for counting processes (100/58); 9. Inference for semimartingale regression models (52/22); 10. Applications to stochastic modeling (10/15).
The first two chapters are introductory. Chapter 1 surveys material from well-known textbooks, in particular about semimartingales, stochastic integration, Girsanov theorems, limit theorems, diffusion-type processes, and point processes. Chapter 2 mainly refers to the theory developed by U. Küchler and M. Sørensen [Int. Stat. Rev. 57, No. 2, 123-144 (1989; Zbl 0721.60055); J. Stat. Plann. Inference 39, No. 2, 211-237 (1994; Zbl 0854.60076)]; however, there is no reference to their recent book, Exponential families of stochastic processes. (1997; Zbl 0882.60012).
Chapters 3-7 are devoted to the (abstract) asymptotic theory, summarizing various previously published results. Chapter 3 treats existence, consistency and asymptotic distributions of likelihood equation estimators both for general processes and a class of exponential families of semimartingales. Besides that, the role of several information quantities is thoroughly discussed. Chapter 4 specifies the asymptotic likelihood theory to diffusions with jumps. Chapter 5 presents the concept of (optimal) estimating functions in applications to semimartingales and partially specified counting processes. Chapter 6 covers the concepts of local asymptotic normality, - mixed normality, - quadraticity, and - infinite divisibility. Chapter 7 discusses asymptotic efficiency, both for models with fully and with partially specified likelihoods. The partial likelihood approach is covered as well. Particular emphasis lies on counting process models.
Chapters 8 and 9 turn to a thorough specific treatment of two classes of models. Chapter 8 deals with parametric, semiparametric, and nonparametric inference for counting processes with particular attention to multiplicative and additive hazard models. Chapter 9 contains a general exposition of semimartingale regression models and treats some particular cases (linear dynamical systems with several types of noise) in detail. It should be mentioned that the class of diffusion (-type) models is not included in this book because the author devoted a separate book to this topic [Statistical inference for diffusion-type processes. (1999; Zbl 0952.62077)].
The short Chapter 10 gives hints to “Applications in stochastic modeling”. Six appendices provide necessary mathematical tools, and the seventh collects bibliographical notes. It might have been desirable to extend the bibliographical information in order to refer to other recent publications related to the topics of this book [e.g., Yu.A. Kutoyants, Statistical inference for spatial Poisson processes. (1998; Zbl 0904.62108); C.C. Heyde, Quasi-likelihood and its application. A general approach to optimal parameter estimation. (1997; Zbl 0879.62076); G. Last and A. Brandt, Marked point processes on the real line. The dynamic approach. (1995; Zbl 0829.60038)].