Mathematical foundations of infinite-dimensional statistical models.

*(English)*Zbl 1358.62014
Cambridge Series in Statistical and Probabilistic Mathematics 40. Cambridge: Cambridge University Press (ISBN 978-1-107-04316-9/hbk; 978-1-107-33786-2/ebook). xiv, 690 p. (2016).

Let us start with saying that this is a very interesting book intended especially for the theoretically oriented statisticians interested in nonparametrical and high-dimensional modelling. For these domains it is well known that the classical Gauss-Fisher-LeCam theory of the optimality of maximum likelihood and Bayesian posterior inference does not apply, so that many new foundations and ideas had to be developed during the past several decades. Both authors contributed to this development considerably.

This monograph presents a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained “mini-courses” on the theory of Gaussian and empirical processes, on approximation and wavelet theory, and on the basic theory of function spaces. The theory of statistical inference in such models – hypothesis testing, estimation and confidence sets – is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, as well as Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski’s method, wavelet thresholding and adaptive confidence regions for self-similar functions.

The presentation of the main statistical material concentrates on function estimation problems. Many other nonparametric models have similar features but are formally different. The aim is to present a unified statistical theory for a canonical family of infinite dimensional models, and this comes at the expense of the breath of topics that could be covered. However, the mathematical mechanism described in the book can serve as guiding principle for many nonparametric problems not covered in this book.

Each chapter is organized in several sections, and historical notes complementing each section can be found at the end of each chapter. At the end of each section exercises are provided. They complement the main results and of the text and often indicate interesting applications or extensions of the material presented in the book.

This monograph presents a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained “mini-courses” on the theory of Gaussian and empirical processes, on approximation and wavelet theory, and on the basic theory of function spaces. The theory of statistical inference in such models – hypothesis testing, estimation and confidence sets – is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, as well as Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski’s method, wavelet thresholding and adaptive confidence regions for self-similar functions.

The presentation of the main statistical material concentrates on function estimation problems. Many other nonparametric models have similar features but are formally different. The aim is to present a unified statistical theory for a canonical family of infinite dimensional models, and this comes at the expense of the breath of topics that could be covered. However, the mathematical mechanism described in the book can serve as guiding principle for many nonparametric problems not covered in this book.

Each chapter is organized in several sections, and historical notes complementing each section can be found at the end of each chapter. At the end of each section exercises are provided. They complement the main results and of the text and often indicate interesting applications or extensions of the material presented in the book.

Reviewer: Jaromír Antoch (Praha)

##### MSC:

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

62G07 | Density estimation |

62C10 | Bayesian problems; characterization of Bayes procedures |

62C12 | Empirical decision procedures; empirical Bayes procedures |

62C20 | Minimax procedures in statistical decision theory |

62G05 | Nonparametric estimation |

62G20 | Asymptotic properties of nonparametric inference |

42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |

62-03 | History of statistics |

01A60 | History of mathematics in the 20th century |

62G08 | Nonparametric regression and quantile regression |

60G15 | Gaussian processes |

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |