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Mathematical Foundations of Infinite-Dimensional Statistical Models
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.
Fundamentals of Nonparametric Bayesian Inference
Bayesian nonparametrics comes of age with this landmark text synthesizing theory, methodology and computation.
High-Dimensional Probability
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
String Theory and the Scientific Method
Explains why string theorists develop a strong belief in their theory despite the lack of empirical confirmation.
The Fundamentals of Heavy Tails
An accessible yet rigorous package of probabilistic and statistical tools for anyone who must understand or model extreme events.
Random Graphs and Complex Networks
The definitive introduction to the local and global structure of random graph models for complex networks.
