渐近统计

本书是一部介绍渐近统计经典教材，内容实用而且数学理论论述严谨。书中除了介绍介绍渐近统计的核心内容——似然推断，M估计，渐近效率，U统计和秩过程，书中还涉及该领域的新研究论题，如半参数模型，自助法，经验过程，等其他应用。本书各章有习题。

Preface Notation 1. Introduction 1.1. Approximate Statistical Procedures 1.2. Asymptotic Optimality Theory 1.3. Limitations 1.4. The Index n 2. Stochastic Convergence 2.1. Basic Theory 2.2. Stochastic o and O Symbols 2.3. Characteristic Functions 2.4. Almost-Sure Representations 2.5. Convergence of Moments 2.6. Convergence-Determining Classes 2.7. Law of the Iterated Logarithm 2.8. Lindeberg-Feller Theorem 2.9. Convergence in Total Variation Problems 3. Delta Method 3.1. Basic Result 3.2. Variance-Stabilizing Transformations 3.3. Higher-Order Expansions 3.4. Uniform Delta Method 3.5. Moments Problems 4. Moment Estimators 4.1. Method of Moments 4.2. Exponential Families Problems 5. M- and Z-Estimators 5.1. Introduction 5.2. Consistency 5.3. Asymptotic Normality 5.4. Estimated Parameters 5.5. Maximum Likelihood Estimators 5.6. Classical Conditions 5.7. One-Step Estimators 5.8. Rates of Convergence 5.9. Argmax Theorem Problems 6. Contiguity 6.1. Likelihood Ratios 6.2. Contiguity Problems 7. Local Asymptotic Normality 7.1. Introduction 7.2. Expanding the Likelihood 7.3. Convergence to a Normal Experiment 7.4. Maximum Likelihood 7.5. Limit Distributions under Alternatives 7.6. Local Asymptotic Normality Problems 8. Efficiency of Estimators 8.1. Asymptotic Concentration 8,2, Relative Efficiency 8.3. Lower Bound for Experiments 8.4. Estimating Normal Means 8.5. Convolution Theorem 8.6. Almost-Everywhere Convolution Theorem 8.7. Local Asymptotic Minimax Theorem 8.8. Shrinkage Estimators 8.9. Achieving the Bound 8.10. Large Deviations Problems 9. Limits of Experiments 9.1. Introduction 9.2. Asymptotic Representation Theorem 9.3. Asymptotic Normality 9.4. Uniform Distribution 9.5. Pareto Distribution 9.6. Asymptotic Mixed Normality 9.7. Heuristics Problems 10. Bayes Procedures 10.1. Introduction 10.2. Bernstein-von Mises Theorem 10.3. Point Estimators 10.4. Consistency Problems 11. Projections 11.1. Projections 11.2. Conditional Expectation 11.3. Projection onto Sums 11.4. Hoeffding Decomposition Problems 12. U-Statistics 12.1. One-Sample U-Statistics 12.2. Two-Sample U-statistics 12.3. Degenerate U-Statistics Problems 13. Rank, Sign, and Permutation Statistics 13.1. Rank Statistics 13.2. Signed Rank Statistics 13.3. Rank Statistics for Independence 13.4. Rank Statistics under Alternatives 13.5. Permutation Tests 13.6. Rank Central Limit Theorem Problems 14. Relative Efficiency of Tests 14.1. Asymptotic Power Functions 14.2. Consistency 14.3. Asymptotic Relative Efficiency 14.4. Other Relative Efficiencies 14.5. Rescaling Rates Problems 15. Efficiency of Tests 15.1. Asymptotic Representation Theorem 15.2. Testing Normal Means 15.3. Local Asymptotic Normality 15.4. One-Sample Location 15.5. Two-Sample Problems Problems 16. Likelihood Ratio Tests 16.1. Introduction 16.2. Taylor Expansion 16.3. Using Local Asymptotic Normality 16.4. Asymptotic Power Functions 16.5. Bartlett Correction 16.6. Bahadur Efficiency Problems 17. Chi-Square Tests 17.1. Quadratic Forms in Normal Vectors 17.2. Pearson Statistic 17.3. Estimated Parameters 17.4. Testing Independence 17.5. Goodness-of-Fit Tests 17.6. Asymptotic Efficiency Problems 18. Stochastic Convergence in Metric Spaces 18.1. Metric and Normed Spaces 18.2. Basic Properties 18.3. Bounded Stochastic Processes Problems 19. Empirical Processes 19.1. Empirical Distribution Functions 19.2. Empirical Distributions 19.3. Goodness-of-Fit Statistics 19.4. Random Functions 19.5. Changing Classes 19.6. Maximal Inequalities Problems 20. Functional Delta Method 20.1. yon Mises Calculus 20.2. Hadamard-Differentiable Functions 20.3. Some Examples Problems 21. Quantiles and Order Statistics 21.1. Weak Consistency 21.2. Asymptotic Normality 21.3. Median Absolute Deviation 21.4. Extreme Values Problems 22. L-Statistics 22.1. Introduction 22.2. Hajek Projection 22.3. Delta Method 22.4. L-Estimators for Location Problems 23. Bootstrap 23.1. Introduction 23.2. Consistency 23.3. Higher-Order Correctness Problems 24. Nonparametric Density Estimation 24.1 Introduction 24.2 Kernel Estimators 24.3 Rate Optimality 24.4 Estimating a Unimodal Density Problems 25. Semiparametric Models 25.1 Introduction 25.2 Banach and Hilbert Spaces 25.3 Tangent Spaces and Information 25.4 Efficient Score Functions 25.5 Score and Information Operators 25.6 Testing 25.7 Efficiency and the Delta Method 25.8 Efficient Score Equations 25.9 General Estimating Equations 25.10 Maximum Likelihood Estimators 25.11 Approximately Least-Favorable Submodels 25.12 Likelihood Equations Problems References Index

A. W. van der Vaart (A.W.范德瓦特,荷兰)有多部著作，本书是其代表作之一，还著有Weak Convergence and Empirical Processes: With Applications to Statistics。

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