From Finite Sample to Asymptotic Methods in

Exact statistical inference may be employed in diverse fields of science and technology. As problems become more complex and sample sizes become larger, mathematical and computational difficulties can arise that require the use of approximate statistical methods. Such methods are justified by asymptotic arguments but are still based on the concepts and principles that underlie exact statistical inference. With this in perspective, this book presents a broad view of exact statistical inference and the development of asymptotic statistical inference, providing a justification for the use of asymptotic methods for large samples. Methodological results are developed on a concrete and yet rigorous mathematical level and are applied to a variety of problems that include categorical data, regression, and survival analyses. This book is designed as a textbook for advanced undergraduate or beginning graduate students in statistics, biostatistics, or applied statistics but may also be used as a reference for academic researchers.

Pranab K. Sen is the Cary C. Boshamer Professor of Biostatistics and Professor of Statistics and Operations Research at the University of North Carolina at Chapel Hill. Julio M. Singer is a Professor in the Department of Statistics, University of Satild;o Paulo, Brazil, and is the codirector of the University's Center for Applied Statistics. Antonio C. Pedroso de Lima is an Associate Professor in the Department of Statistics, University of Satild;o Paulo, Brazil, and is the codirector of the university's Center for Applied Statistics.

Preface	p. xi
Motivation and Basic Tools	p. 1
Introduction	p. 1
Illustrative Examples and Motivation	p. 2
Synthesis of Finite to Asymptotic Statistical Methods	p. 8
The Organization of the Book	p. 13
Basic Tools and Concepts	p. 15
Exercises	p. 38
Estimation Theory	p. 42
Introduction	p. 42
Basic Concepts	p. 42
Likelihood, Information, and Sufficiency	p. 45
Methods of Estimation	p. 55
Finite Sample Optimality Perspectives	p. 62
Concluding Notes	p. 65
Exercises	p. 66
Hypothesis Testing	p. 68
Introduction	p. 68
The Neyman-Pearson Paradigm	p. 68
Composite Hypotheses: Beyond the Neyman-Pearson Paradigm	p. 73
Invariant Tests	p. 80
Concluding Notes	p. 81
Exercises	p. 81
Elements of Statistical Decision Theory	p. 83
Introduction	p. 83
Basic Concepts	p. 83
Bayes Estimation Methods	p. 88
Bayes Hypothesis Testing	p. 93
Confidence Sets	p. 95
Concluding Notes	p. 97
Exercises	p. 97
Stochastic Processes: An Overview	p. 100
Introduction	p. 100
Processes with Markov Dependencies	p. 102
Discrete Time-Parameter Processes	p. 110
Continuous Time-Parameter Processes	p. 112
Exercises	p. 118
Stochastic Convergence and Probability Inequalities	p. 119
Introduction	p. 119
Modes of Stochastic Convergence	p. 121
Probability Inequalities and Laws of Large Numbers	p. 131
Extensions to Dependent Variables	p. 160
Miscellaneous Convergence Results	p. 164
Concluding Notes	p. 169
Exercises	p. 169
Asymptotic Distributions	p. 173
Introduction	p. 173
Some Important Tools	p. 177
Central Limit Theorems	p. 181
Rates of Convergence to Normality	p. 197
Projections and Variance-Stabilizing Transformations	p. 201
Quadratic Forms	p. 218
Order Statistics and Empirical Distributions	p. 221
Concluding Notes	p. 232
Exercises	p. 236
Asymptotic Behavior of Estimators and Tests	p. 240
Introduction	p. 240
Estimating Equations and Local Asymptotic Linearity	p. 240
Asymptotics for MLE	p. 245
Asymptotics for Other Classes of Estimators	p. 249
Asymptotic Efficiency of Estimators	p. 255
Asymptotic Behavior of Some Test Statistics	p. 259
Resampling Methods	p. 268
Concluding Remarks	p. 271
Exercises	p. 272
Categorical Data Models	p. 273
Introduction	p. 273
Nonparametric Goodness-of-Fit Tests	p. 275
Estimation and Goodness-of-Fit Tests: Parametric Case	p. 278
Some Other Important Statistics	p. 286
Concluding Notes	p. 288
Exercises	p. 289
Regression Models	p. 293
Introduction	p. 293
Generalized Least-Squares Procedures	p. 295
Robust Estimators	p. 308
Nonlinear Regression Models	p. 316
Generalized Linear Models	p. 317
Generalized Least-Squares Versus Generalized Estimating Equations	p. 328
Nonparametric Regression	p. 331
Concluding Notes	p. 335
Exercises	p. 336
Weak Convergence and Gaussian Processes	p. 338
Introduction	p. 338
Weak Invariance Principles	p. 338
Weak Convergence of Partial Sum Processes	p. 341
Weak Convergence of Empirical Processes	p. 350
Weak Convergence and Statistical Functionals	p. 360
Weak Convergence and Nonparametrics	p. 365
Strong Invariance Principles	p. 371
Concluding Notes	p. 372
Exercises	p. 373
Bibliography	p. 375
Index	p. 381
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From Finite Sample to Asymptotic Methods in Statistics

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