Mathematical Foundations of Infinite-Dimensional Statistical Models

Now in paperback: the new classic on the theory of statistical inference in statistical models with an infinite-dimensional parameter space.

Evarist Giné (Author), Richard Nickl (Author)

9781108994132, Cambridge University Press

Paperback / softback, published 25 March 2021

704 pages
25.1 x 17.6 x 3.6 cm, 1.27 kg

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.

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.

Subject Areas: Signal processing [UYS], Machine learning [UYQM], Probability & statistics [PBT], Calculus & mathematical analysis [PBK], Econometrics [KCH]