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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.
String Theory and the Scientific Method
Explains why string theorists develop a strong belief in their theory despite the lack of empirical confirmation.
Random Graphs and Complex Networks
The definitive introduction to the local and global structure of random graph models for complex networks.
The Fundamentals of Heavy Tails
An accessible yet rigorous package of probabilistic and statistical tools for anyone who must understand or model extreme events.
Probability
A well-written and lively introduction to measure theoretic probability for graduate students and researchers.
Regularity and Irregularity of Superprocesses with (1 + β)-stable Branching Mechanism
This is the only book discussing multifractal properties of densities of stable superprocesses, containing latest achievements while also giving the reader a comprehensive picture of the state of the art in this area. It is a self-contained presentation of regularity properties of stable superprocesses and proofs of main results and can serve as an introductory text for a graduate course. There are many heuristic explanations of technically involved results and proofs and the reader can get a clear intuitive picture behind the results and techniques.
Principles of Statistical Analysis
This concise course in data analysis and inference for the mathematically literate builds on survey sampling and designed experiments.
