Point processes and their statistical inference. 2nd ed., rev. and expanded.

*(English)*Zbl 0733.62088
Probability: Pure and Applied, 7, 7. New York etc.: Marcel Dekker, Inc. xiv, 490 p. $ 110.00 (US and Canada); $ 126.00 (all other countries) (1991).

Without doubt this second (revised and expanded) edition of the author’s book will find the same good resonance as the first, 1986-edition did; see the review Zbl 0601.62120. Statistics of point processes is a branch of both probability theory and mathematical statistics with steadily increasing importance in several areas of application such as econometrics, biometrics, medicine and environmental sciences. The present book, however, is primarily a research monograph intended for mathematicians. Nevertheless it is suitable as a text for graduate-level courses and seminars.

The first two chapters introduce into two fundamental aspects of point processes which, at first sight, are rather distinct: the distribution theory and the intensity theory. In connection with the distributed approach to point processes, such topics are discussed as the relationship with random measures, marked point processes and Palm distributions, whereas the intensity-based approach discusses the use of martingale methods, in particular compensators, stochastic intensities and the representation of martingales driven by point processes.

The next chapter introduces into some basic concepts of statistical inference and state estimation. For giving an example, statistics of Poisson processes are considered in detail. Chapter 4 is devoted to empirical measures as estimators of point-process distributions on a general compact space, to their asymptotic properties and to estimation of functionals of point processes based on corresponding empirical functionals such as empirical Laplace functionals and empirical Palm distributions. In Chapter 5, the martingale method of inference is considered for point processes on the non-negative halfline which admit a stochastic intensity. In connection with this, large-sample properties of martingale estimators are investigated. For the multiplicative intensity model, maximum likelihood estimators are considered, as well as hypothesis testing. The Cox regression model and several extensions are also discussed.

Returning to the case of a general state space, the next two chapters are concerned with inference for special classes of point processes exploiting specific structural assumptions. In particular, estimation and hypothesis testing for Poisson and Cox processes is considered. Chapters 8 and 9 deal with nonparametric inference for renewal processes and for Markov renewal processes on the real line as well as for stationary point processes in the d-dimensional Euclidean space. In a final chapter, the attention is focused on inference for stochastic processes based on Poisson process samples.

In comparison with the first edition, the sections concerning the multiplicative intensity model and stationary point processes have been completely reorganized. New material has been added for the Cox regression model. Many fundamental statistical concepts are actually explained in more details. Furthermore, the bibliography has been expanded and updated. At the end of each chapter, notes and exercises are given. The book has two appendices providing necessary background on spaces of measures and continuous-time martingales.

The first two chapters introduce into two fundamental aspects of point processes which, at first sight, are rather distinct: the distribution theory and the intensity theory. In connection with the distributed approach to point processes, such topics are discussed as the relationship with random measures, marked point processes and Palm distributions, whereas the intensity-based approach discusses the use of martingale methods, in particular compensators, stochastic intensities and the representation of martingales driven by point processes.

The next chapter introduces into some basic concepts of statistical inference and state estimation. For giving an example, statistics of Poisson processes are considered in detail. Chapter 4 is devoted to empirical measures as estimators of point-process distributions on a general compact space, to their asymptotic properties and to estimation of functionals of point processes based on corresponding empirical functionals such as empirical Laplace functionals and empirical Palm distributions. In Chapter 5, the martingale method of inference is considered for point processes on the non-negative halfline which admit a stochastic intensity. In connection with this, large-sample properties of martingale estimators are investigated. For the multiplicative intensity model, maximum likelihood estimators are considered, as well as hypothesis testing. The Cox regression model and several extensions are also discussed.

Returning to the case of a general state space, the next two chapters are concerned with inference for special classes of point processes exploiting specific structural assumptions. In particular, estimation and hypothesis testing for Poisson and Cox processes is considered. Chapters 8 and 9 deal with nonparametric inference for renewal processes and for Markov renewal processes on the real line as well as for stationary point processes in the d-dimensional Euclidean space. In a final chapter, the attention is focused on inference for stochastic processes based on Poisson process samples.

In comparison with the first edition, the sections concerning the multiplicative intensity model and stationary point processes have been completely reorganized. New material has been added for the Cox regression model. Many fundamental statistical concepts are actually explained in more details. Furthermore, the bibliography has been expanded and updated. At the end of each chapter, notes and exercises are given. The book has two appendices providing necessary background on spaces of measures and continuous-time martingales.

Reviewer: V.Schmidt (Freiberg)

##### MSC:

62M09 | Non-Markovian processes: estimation |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

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

62M30 | Inference from spatial processes |

62M99 | Inference from stochastic processes |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60G44 | Martingales with continuous parameter |