Mathematical Foundations of Infinite-Dimensional Statistical Models

Erscheinungsjahr 2021
ISBN: 978-1-108-99413-2
Verlag: Cambridge University Press

Buch, Englisch, Band 40, 720 Seiten, Format (B × H): 178 mm x 254 mm, Gewicht: 1304 g

Reihe: Cambridge Series in Statistical and Probabilistic Mathematics

In nonparametric and high-dimensional statistical models, the classical Gauss–Fisher–Le Cam theory of the optimality of maximum likelihood estimators and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives 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, approximation and wavelet theory, and the basic theory of function spaces. The theory of statistical inference in such models - hypothesis testing, estimation and confidence sets - is presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, but also Bayesian nonparametrics and nonparametric maximum likelihood estimation. In a final chapter the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding, and adaptive inference for self-similar functions. Winner of the 2017 PROSE Award for Mathematics.

Autoren/Hrsg.

Giné, Evarist

Nickl, Richard

Fachgebiete

• Mathematik | Informatik EDV | Informatik Informatik Künstliche Intelligenz Maschinelles Lernen
• Mathematik | Informatik Mathematik Mathematische Analysis
• Mathematik | Informatik Mathematik Stochastik Wahrscheinlichkeitsrechnung
• Technische Wissenschaften Sonstige Technologien | Angewandte Technik Signalverarbeitung, Bildverarbeitung, Scanning
• Wirtschaftswissenschaften Volkswirtschaftslehre Volkswirtschaftslehre Allgemein Ökonometrie
• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik

Inhaltsverzeichnis

Preface; 1. Nonparametric statistical models; 2. Gaussian processes; 3. Empirical processes; 4. Function spaces and approximation theory; 5. Linear nonparametric estimators; 6. The minimax paradigm; 7. Likelihood-based procedures; 8. Adaptive inference; References; Author Index; Index.

Über Autor(innen)

Nickl, Richard
Richard Nickl is Professor of Mathematical Statistics in the Statistical Laboratory within the Department of Pure Mathematics and Mathematical Statistics at the University of Cambridge.

Giné, Evarist
Evarist Giné (1944–2015) was Head of the Department of Mathematics at the University of Connecticut. Giné was a distinguished mathematician who worked on mathematical statistics and probability in infinite dimensions. He was the author of two books and more than 100 articles.