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| Author | : Evarist Giné |
| Publisher | : Cambridge University Press |
| Total Pages | : 705 |
| Release | : 2016 |
| Genre | : Business & Economics |
| ISBN | : 1107043166 |
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.
| Author | : Evarist Giné |
| Publisher | : Cambridge University Press |
| Total Pages | : 705 |
| Release | : 2015-11-18 |
| Genre | : Mathematics |
| ISBN | : 1316445178 |
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.
| Author | : Evarist Giné |
| Publisher | : Cambridge University Press |
| Total Pages | : 706 |
| Release | : 2021-03-25 |
| Genre | : Mathematics |
| ISBN | : 1009022784 |
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.
| Author | : David Williams |
| Publisher | : Cambridge University Press |
| Total Pages | : 274 |
| Release | : 1991-02-14 |
| Genre | : Mathematics |
| ISBN | : 9780521406055 |
This is a masterly introduction to the modern, and rigorous, theory of probability. The author emphasises martingales and develops all the necessary measure theory.
| Author | : Olav Kallenberg |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 536 |
| Release | : 2005-07-27 |
| Genre | : Mathematics |
| ISBN | : 9780387251158 |
This is the first comprehensive treatment of the three basic symmetries of probability theory—contractability, exchangeability, and rotatability—defined as invariance in distribution under contractions, permutations, and rotations. Originating with the pioneering work of de Finetti from the 1930's, the theory has evolved into a unique body of deep, beautiful, and often surprising results, comprising the basic representations and invariance properties in one and several dimensions, and exhibiting some unexpected links between the various symmetries as well as to many other areas of modern probability. Most chapters require only some basic, graduate level probability theory, and should be accessible to any serious researchers and graduate students in probability and statistics. Parts of the book may also be of interest to pure and applied mathematicians in other areas. The exposition is formally self-contained, with detailed references provided for any deeper facts from real analysis or probability used in the book. Olav Kallenberg received his Ph.D. in 1972 from Chalmers University in Gothenburg, Sweden. After teaching for many years at Swedish universities, he moved in 1985 to the US, where he is currently Professor of Mathematics at Auburn University. He is well known for his previous books Random Measures (4th edition, 1986) and Foundations of Modern Probability (2nd edition, 2002) and for numerous research papers in all areas of probability. In 1977, he was the second recipient ever of the prestigious Rollo Davidson Prize from Cambridge University. In 1991–94, he served as the Editor in Chief of Probability Theory and Related Fields. Professor Kallenberg is an elected fellow of the Institute of Mathematical Statistics.
| Author | : Jianqing Fan |
| Publisher | : CRC Press |
| Total Pages | : 942 |
| Release | : 2020-09-21 |
| Genre | : Mathematics |
| ISBN | : 0429527616 |
Statistical Foundations of Data Science gives a thorough introduction to commonly used statistical models, contemporary statistical machine learning techniques and algorithms, along with their mathematical insights and statistical theories. It aims to serve as a graduate-level textbook and a research monograph on high-dimensional statistics, sparsity and covariance learning, machine learning, and statistical inference. It includes ample exercises that involve both theoretical studies as well as empirical applications. The book begins with an introduction to the stylized features of big data and their impacts on statistical analysis. It then introduces multiple linear regression and expands the techniques of model building via nonparametric regression and kernel tricks. It provides a comprehensive account on sparsity explorations and model selections for multiple regression, generalized linear models, quantile regression, robust regression, hazards regression, among others. High-dimensional inference is also thoroughly addressed and so is feature screening. The book also provides a comprehensive account on high-dimensional covariance estimation, learning latent factors and hidden structures, as well as their applications to statistical estimation, inference, prediction and machine learning problems. It also introduces thoroughly statistical machine learning theory and methods for classification, clustering, and prediction. These include CART, random forests, boosting, support vector machines, clustering algorithms, sparse PCA, and deep learning.
| Author | : Inge Koch |
| Publisher | : Cambridge University Press |
| Total Pages | : 531 |
| Release | : 2014 |
| Genre | : Business & Economics |
| ISBN | : 0521887933 |
This modern approach integrates classical and contemporary methods, fusing theory and practice and bridging the gap to statistical learning.
| Author | : Subhashis Ghosal |
| Publisher | : Cambridge University Press |
| Total Pages | : 671 |
| Release | : 2017-06-26 |
| Genre | : Business & Economics |
| ISBN | : 0521878268 |
Bayesian nonparametrics comes of age with this landmark text synthesizing theory, methodology and computation.
| Author | : Roman Vershynin |
| Publisher | : Cambridge University Press |
| Total Pages | : 299 |
| Release | : 2018-09-27 |
| Genre | : Business & Economics |
| ISBN | : 1108415199 |
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
| Author | : Martin J. Wainwright |
| Publisher | : Cambridge University Press |
| Total Pages | : 571 |
| Release | : 2019-02-21 |
| Genre | : Business & Economics |
| ISBN | : 1108498027 |
A coherent introductory text from a groundbreaking researcher, focusing on clarity and motivation to build intuition and understanding.
