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Textbook in PDF format The theory of empirical processes provides valuable tools for the development of asymptotic theory in (nonparametric) statistical models, and makes it possible to give a unified treatment of various models. This book reveals the relation between the asymptotic behavior of M-estimators and the complexity of parameter space, using entropy as a measure of complexity, presenting tools and methods to analyze nonparametric, and in some cases, semiparametric methods. Graduate students and professionals in statistics, as well as those interested in applications, e.g. to econometrics, medical statistics, etc., will welcome this treatment. Preface Guide to the Reader Introduction Some examples from statistics Problems and complements Notation and Definitions Stochastic order symbols The empirical process Entropy Examples Notes Problems and complements Uniform Laws of Large Numbers Uniform laws of large numbers under finite entropy with bracketing The chaining technique A maximal inequality for weighted sums Symmetrization Hoeffding's inequality Uniform laws of large numbers under random entropy conditions Examples Notes Problems and complements First Applications: Consistency Consistency of maximum likelihood estimators Examples Consistency of least squares estimators Examples Notes Problems and complements Increments of Empirical Processes Random entropy numbers and asymptotic equicontinuity Random entropy numbers and classes depending on n Empirical entropy and empirical norms A uniform inequality based on entropy with bracketing Entropy with bracketing and asymptotic equicontinuity Modulus of continuity Entropy with bracketing and empirical norms Notes Problems and complements Central Limit Theorems Definitions Sufficient conditions for 3 to be P-Donsker Useful theorems Measurability Notes Problems and complements Rates of Convergence for Maximum Likelihood Estimators The main idea An exponential inequality for the maximum likelihood estimator Convex classes of densities Examples Notes Problems and complements The Non-I.I.D. Case Independent non-identically distributed random variables Maximal inequalities for weighted sums revisited Martingales Application to maximum likelihood Examples Notes Problems and complements Rates of Convergence for Least Squares Estimators Sub-Gaussian errors Errors with exponential tails Examples Notes Problems and complements Penalties and Sieves Penalized least squares Penalized maximum likelihood Roughness penalty on the density Roughness penalty on the log-density Least squares on sieves Maximum likelihood on sieves Notes Problems and complements Some Applications to Semiparametric Models Partial linear models Mixture models Introduction A single-indexed model with binary explanatory variable Notes Problems and complements M-Estimators Introduction Estimating a regression function using a general loss function Classes of functions indexed by a finite-dimensional parameter Least squares Maximum likelihood Asymptotic normality Notes Problems and complements Appendix References Symbol Index Author Index Subject Index
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