Mathematical Foundations of Infinite-Dimensional Statistical Models

This book develops the theory of statistical inference in statistical models with an infinite-dimensional parameter space, including mathematical foundations and key decision-theoretic principles.

Evarist Giné (Author), Richard Nickl (Author)

9781107043169, Cambridge University Press

Hardback, published 18 November 2015

720 pages
26.1 x 18.6 x 4.5 cm, 1.38 kg

'This is a remarkably comprehensive, detailed and rigorous treatment of mathematical theory for non-parametric and high-dimensional statistics. Special emphasis is on density and regression estimation and corresponding confidence sets and hypothesis testing. The minimax paradigm and adaptivity play a key role.' Natalie Neumeyer, MathSciNet

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, 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, but also 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 inference for self-similar functions.

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.

Subject Areas: Probability & statistics [PBT], Econometrics [KCH], Economics [KC], Economics, finance, business & management [K]