Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

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.

Nonparametric statistics. --- Function spaces.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

For almost fifty years, Richard M. Dudley has been extremely influential in the development of several areas of Probability. His work on Gaussian processes led to the understanding of the basic fact that their sample boundedness and continuity should be characterized in terms of proper measures of complexity of their parameter spaces equipped with the intrinsic covariance metric. His sufficient condition for sample continuity in terms of metric entropy is widely used and was proved by X. Fernique to be necessary for stationary Gaussian processes, whereas its more subtle versions (majorizing measures) were proved by M. Talagrand to be necessary in general. Together with V. N. Vapnik and A. Y. Cervonenkis, R. M. Dudley is a founder of the modern theory of empirical processes in general spaces. His work on uniform central limit theorems (under bracketing entropy conditions and for Vapnik-Cervonenkis classes), greatly extends classical results that go back to A. N. Kolmogorov and M. D. Donsker, and became the starting point of a new line of research, continued in the work of Dudley and others, that developed empirical processes into one of the major tools in mathematical statistics and statistical learning theory. As a consequence of Dudley's early work on weak convergence of probability measures on non-separable metric spaces, the Skorohod topology on the space of regulated right-continuous functions can be replaced, in the study of weak convergence of the empirical distribution function, by the supremum norm. In a further recent step Dudley replaces this norm by the stronger p-variation norms, which then allows replacing compact differentiability of many statistical functionals by Fréchet differentiability in the delta method. Richard M. Dudley has also made important contributions to mathematical statistics, the theory of weak convergence, relativistic Markov processes, differentiability of nonlinear operators and several other areas of mathematics. Professor Dudley has been the adviser to thirty PhD's and is a Professor of Mathematics at the Massachusetts Institute of Technology.

Mathematics. --- Probabilities. --- Statistics. --- Probabilities --- Mathematics --- Physical Sciences & Mathematics --- Mathematical Statistics --- Probability --- Statistical inference --- Probability Theory and Stochastic Processes. --- Statistical Theory and Methods. --- Distribution (Probability theory. --- Mathematical statistics. --- Distribution functions --- Frequency distribution --- Characteristic functions --- Statistics, Mathematical --- Statistics --- Sampling (Statistics) --- Statistical methods --- Statistics . --- Statistical analysis --- Statistical data --- Statistical science --- Econometrics --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Stochastic processes --- Mathematical statistics --- Probabiliteit--Theorie --- Probabiliteitstheorie --- Probabilité [Théorie de la ] --- Probabilités --- Statistieken --- Waarschijnlijkheid--Theorie --- Waarschijnlijkheidstheorie --- Gibbs' equation --- Perturbation (Mathematics) --- Inverse problems (Differential equations) --- Gaussian measures --- Isoperimetric inequalities --- Statistique mathématique --- Problèmes inverses (Equations différentielles) --- Probabilités --- Statistique mathématique --- Problèmes inverses (Equations différentielles) --- Statistiques --- Probabilities --- Congresses. --- Congrès --- Processus stochastiques --- Probabilités. --- Statistique mathématique. --- Probabilités - Congres --- Statistiques - Congres --- Statistique mathématique.