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1.Introduction 1

1.1.Approximate Statistical Procedures 1

1.2.Asymptotic Optimality Theory 2

1.3.Limitations 3

1.4.The Index n 4

2.Stochastic Convergence 5

2.1.Basic Theory 5

2.2.Stochastic o and O Symbols 12

2.3.Characteristic Functions 13

2.4.Almost-Sure Representations 17

2.5.Convergence of Moments 17

2.6.Convergence-Determining Classes 18

2.7.Law of the Iterated Logarithm 19

2.8.Lindeberg-Feller Theorem 20

2.9.Convergence in Total Variation 22

Problems 24

3.Delta Method 25

3.1.Basic Result 25

3.2.Variance-Stabilizing Transformations 30

3.3.Higher-Order Expansions 31

3.4.Uniform Delta Method 32

3.5.Moments 33

Problems 34

4.Moment Estimators 35

4.1.Method of Moments 35

4.2.Exponential Families 37

Problems 40

5.M-and Z-Estimators 41

5.1.Introduction 41

5.2.Consistency 44

5.3.Asymptotic Normality 51

5.4.Estimated Parameters 60

5.5.Maximum Likelihood Estimators 61

5.6.Classical Conditions 67

5.7.One-Step Estimators 71

5.8.Rates of Convergence 75

5.9.Argmax Theorem 79

Problems 83

6.Contiguity 85

6.1.Likelihood Ratios 85

6.2.Contiguity 87

Problems 91

7.Local Asymptotic Normality 92

7.1.Introduction 92

7.2.Expanding the Likelihood 93

7.3.Convergence to a Normal Experiment 97

7.4.Maximum Likelihood 100

7.5.Limit Distributions under Alternatives 103

7.6.Local Asymptotic Normality 103

Problems 106

8.Efficiency of Estimators 108

8.1.Asymptotic Concentration 108

8.2.Relative Efficiency 110

8.3.Lower Bound for Experiments 111

8.4.Estimating Normal Means 112

8.5.Convolution Theorem 115

8.6.Almost-Everywhere Convolution Theorem 115

8.7.Local Asymptotic Minimax Theorem 117

8.8.Shrinkage Estimators 119

8.9.Achieving the Bound 120

8.10.Large Deviations 122

Problems 123

9.Limits of Experiments 125

9.1.Introduction 125

9.2.Asymptotic Representation Theorem 126

9.3.Asymptotic Normality 127

9.4.Uniform Distribution 129

9.5.Pareto Distribution 130

9.6.Asymptotic Mixed Normality 131

9.7.Heuristics 136

Problems 137

10.Bayes Procedures 138

10.1.Introduction 138

10.2.Bernstein-von Mises Theorem 140

10.3.Point Estimators 146

10.4.Consistency 149

Problems 152

11.Projections 153

11.1.Projections 153

11.2.Conditional Expectation 155

11.3.Projection onto Sums 157

11.4.Hoeffding Decomposition 157

Problems 160

12.U-Statistics 161

12.1.One-Sample U-Statistics 161

12.2.Two-Sample U-statistics 165

12.3.Degenerate U-Statistics 167

Problems 171

13.Rank，Sign，and Permutation Statistics 173

13.1.Rank Statistics 173

13.2.Signed Rank Statistics 181

13.3.Rank Statistics for Independence 184

13.4.Rank Statistics under Alternatives 184

13.5.Permutation Tests 188

13.6.Rank Central Limit Theorem 190

Problems 190

14.Relative Efficiency of Tests 192

14.1.Asymptotic Power Functions 192

14.2.Consistency 199

14.3.Asymptotic Relative Efficiency 201

14.4.Other Relative Efficiencies 202

14.5.Rescaling Rates 211

Problems 213

15.Efficiency of Tests 215

15.1.Asymptotic Representation Theorem 215

15.2.Testing Normal Means 216

15.3.Local Asymptotic Normality 218

15.4.One-Sample Location 220

15.5.Two-Sample Problems 223

Problems 226

16.Likelihood Ratio Tests 227

16.1.Introduction 227

16.2.Taylor Expansion 229

16.3.Using Local Asymptotic Normality 231

16.4.Asymptotic Power Functions 236

16.5.Bartlett Correction 238

16.6.BahadurEfficiency 238

Problems 241

17.Chi-Square Tests 242

17.1.Quadratic Forms in Normal Vectors 242

17.2.Pearson Statistic 242

17.3.Estimated Parameters 244

17.4.Testing Independence 247

17.5.Goodness-of-Fit Tests 248

17.6.Asymptotic Efficiency 251

Problems 253

18.Stochastic Convergence in Metric Spaces 255

18.1.Metric and Normed Spaces 255

18.2.Basic Properties 258

18.3.Bounded Stochastic Processes 260

Problems 263

19.Empirical Processes 265

19.1.Empirical Distribution Functions 265

19.2.Empirical Distributions 269

19.3.Goodness-of-Fit Statistics 277

19.4.Random Functions 279

19.5.Changing Classes 282

19.6.Maximal Inequalities 284

Problems 289

20.Functional Delta Method 291

20.1.von Mises Calculus 291

20.2.Hadamard-Differentiable Functions 296

20.3.Some Examples 298

Problems 303

21.Quantiles and Order Statistics 304

21.1.Weak Consistency 304

21.2.Asymptotic Normality 305

21.3.Median Absolute Deviation 310

21.4.Extreme Values 312

Problems 315

22.L-Statistics 316

22.1.Introduction 316

22.2.Hajek Projection 318

22.3.Delta Method 320

22.4.L-Estimators for Location 323

Problems 324

23.Bootstrap 326

23.1.Introduction 326

23.2.Consistency 329

23.3.Higher-Order Correctness 334

Problems 339

24.Nonparametric Density Estimation 341

24.1 Introduction 341

24.2 Kernel Estimators 341

24.3 Rate Optimality 346

24.4 Estimating a Unimodal Density 349

Problems 356

25.Semiparametric Models 358

25.1 Introduction 358

25.2 Banach and Hilbert Spaces 360

25.3 Tangent Spaces and Information 362

25.4 Efficient Score Functions 368

25.5 Score and Information Operators 371

25.6 Testing 384

25.7 Efficiency and the Delta Method 386

25.8 Efficient Score Equations 391

25.9 General Estimating Equations 400

25.10 Maximum Likelihood Estimators 402

25.11 Approximately Least-Favorable Submodels 408

25.12 Likelihood Equations 419

Problems 431

References 433

Index 439

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