Multivariate Nonparametric Regression and Visualization

Preface xvii

Introduction xix

I.1 Estimation of Functionals of Conditional Distributions xx

I.2 Quantitative Finance xxi

I.3 Visualization xxi

I.4 Literature xxiii

PART I METHODS OF REGRESSION AND CLASSIFICATION

1 Overview of Regression and Classification 3

1.1 Regression 3

1.2 Discrete Response Variable 29

1.3 Parametric Family Regression 33

1.4 Classification 37

1.5 Applications in Quantitative Finance 42

1.6 Data Examples 52

1.7 Data Transformations 53

1.8 Central Limit Theorems 58

1.9 Measuring the Performance of Estimators 61

1.10 Confidence Sets 73

1.11 Testing 75

2 Linear Methods and Extensions 77

2.1 Linear Regression 78

2.2 Varying Coefficient Linear Regression 97

2.3 Generalized Linear and Related Models 102

2.4 Series Estimators 107

2.5 Conditional Variance and ARCH models 111

2.6 Applications in Volatility and Quantile Estimation 115

2.7 Linear Classifiers 124

3 Kernel Methods and Extensions 127

3.1 Regressogram 129

3.2 Kernel Estimator 130

3.3 Nearest Neighborhood Estimator 147

3.4 Classification with Local Averaging 148

3.5 Median Smoothing 151

3.6 Conditional Density Estimators 152

3.7 Conditional Distribution Function Estimation 158

3.8 Conditional Quantile Estimation 160

3.9 Conditional Variance Estimation 162

3.10 Conditional Covariance Estimation 176

3.11 Applications in Risk Management 181

3.12 Applications in Portfolio Selection 205

4 Semiparametric and Structural Models 229

4.1 Single Index Model 230

4.2 Additive Model 234

4.3 Other Semiparametric Models 237

5 Empirical Risk Minimization 241

5.1 Empirical Risk 243

5.2 Local Empirical Risk 247

5.3 Support Vector Machines 257

5.4 Stagewise Methods 259

5.5 Adaptive Regressograms 264

PART II VISUALIZATION

6 Visualization of Data 277

6.1 Scatter Plots 278

6.2 Histogram and Kernel Density Estimator 282

6.3 Dimension Reduction 284

6.4 Observations as Objects 288

7 Visualization of Functions 295

7.1 Slices 296

7.2 Partial Dependence Functions 296

7.3 Reconstruction of Sets 299

7.4 Level Set Trees 303

7.5 Unimodal Densities 326

7.5.1 Probability Content of Level Sets 327

7.5.2 Set Visualization 328

Appendix A: R Tutorial 329

A.1 Data Visualization 329

A.2 Linear Regression 331

A.3 Kernel Regression 332

A.4 Local Linear Regression 341

A.5 Additive Models: Backfitting 344

A.6 Single Index Regression 345

A.7 Forward Stagewise Modeling 347

A.8 Quantile Regression 349

References 351

Author Index 361

Topic Index 365