Applied Nonparametric Regression

433 Pages·1992·4.509 MB·English

by  Wolfgang Härdle

Applied
Nonparametric
Regression
Wolfgang H¨ardle
Humboldt-Universita¨t zu Berlin
Wirtschaftswissenschaftliche Fakulta¨t
Institut fu¨r Statistik und O¨konometrie
Spandauer Str. 1
D–10178 Berlin
1994

Fu¨r Renate Nora Viola Adrian

Contents
I Regression smoothing 1
1 Introduction 3
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Scope of this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Basic idea of smoothing 17
2.1 The stochastic nature of the observations . . . . . . . . . . . . . . . . . . . . 26
2.2 Hurdles for the smoothing process . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Smoothing techniques 31
3.1 Kernel Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Proof of Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.4 k-nearest neighbor estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Orthogonal series estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.6 Spline smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.8 An overview of various smoothers . . . . . . . . . . . . . . . . . . . . . . . . 78
3.9 Recursive techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.10 The regressogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.11 A comparison of kernel, k-NN and spline smoothers . . . . . . . . . . . . . . 87
II The kernel method 111
4 How close is the smooth to the true curve? 113
4.1 The speed at which the smooth curve converges . . . . . . . . . . . . . . . . 116
4.2 Pointwise confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.3 Variability bands for functions . . . . . . . . . . . . . . . . . . . . . . . . . . 139
4.4 Behavior at the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.5 The accuracy as a function of the kernel . . . . . . . . . . . . . . . . . . . . 162
4.6 Bias reduction techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5 Choosing the smoothing parameter 179
5.1 Cross-validation, penalizing functions and the plug-in method. . . . . . . . . 180
5.2 Which selector should be used? . . . . . . . . . . . . . . . . . . . . . . . . . 200
5.3 Local adaptation of the smoothing parameter . . . . . . . . . . . . . . . . . 214
5.4 Comparing bandwidths between laboratories (canonical kernels) . . . . . . . 223
6 Data sets with outliers 229
6.1 Resistant smoothing techniques . . . . . . . . . . . . . . . . . . . . . . . . . 231
6.2 Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7 Nonparametric regression techniques for time series 245
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
7.2 Nonparametric time series analysis . . . . . . . . . . . . . . . . . . . . . . . 247
7.3 Smoothing with dependent errors . . . . . . . . . . . . . . . . . . . . . . . . 263
7.4 Conditional heteroscedastic autoregressive nonlinear models . . . . . . . . . 267
8 Looking for special features and qualitative smoothing 281
8.1 Monotonic and unimodal smoothing . . . . . . . . . . . . . . . . . . . . . . . 282
8.2 Estimation of Zeros and Extrema . . . . . . . . . . . . . . . . . . . . . . . . 291
9 Incorporating parametric components 299
9.1 Partial linear models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
9.2 Shape-invariant modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
9.3 Comparing nonparametric and parametric curves . . . . . . . . . . . . . . . 313
III Smoothing in high dimensions 325
10 Investigating multiple regression by additive models 327
10.1 Regression trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
10.2 Projection pursuit regression . . . . . . . . . . . . . . . . . . . . . . . . . . . 337
10.3 Alternating conditional expectations . . . . . . . . . . . . . . . . . . . . . . 341
10.4 Average derivative estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 348
10.5 Generalized additive models . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
A XploRe 365
A.1 Using XploRe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
A.2 Quantlet Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
A.3 Getting Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
A.4 Basic XploRe Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
B Tables 387
Bibliography 391
Index 407
List of Figures
1.1 Potatoes versus net income. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 potatoes versus net income . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Human height growth versus age. . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Net income densities over time. . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Net income densities over time. . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Temperature response function for Georgia. . . . . . . . . . . . . . . . . . . 12
1.7 Nonparametric flow probability for the St. Mary’s river. . . . . . . . . . . . 13
1.8 Side inpact data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1 Food versus net income. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Food versus net income . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Height versus age. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 potatoes versus net income . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Potatoes versus net income . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 The Epanechnikov kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 The effective kernel weights.... . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3 Local parabolic fits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 First and second derivatives of kernel smoothers. . . . . . . . . . . . . . . . . 44
3.5 Title . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Titel! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 The effective weight function . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.8 amount of sugar in sugar-beet as a function of temperature . . . . . . . . . 68
3.9 A spline smooth of the Motorcycle data set . . . . . . . . . . . . . . . . . . . 72
3.10 Spline smooth with cubic polynomial fit . . . . . . . . . . . . . . . . . . . . 95
3.11 The effective spline kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.12 Equivalent kernel function for the temperature . . . . . . . . . . . . . . . . . 97
3.13 Equivalent kernel function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.14 Huber’s approximation to the effective weight function . . . . . . . . . . . . 99
3.15 A regressogram smooth of the motorcycle data. . . . . . . . . . . . . . . . . 100
3.16 Running median and a - smooth. . . . . . . . . . . . . . . . . . . . . . . 101
k NN
3.17 A kernel smooth applied to a sawtooth function. . . . . . . . . . . . . . . . . 102
3.18 The split linear fit applied to a sawtooth function. . . . . . . . . . . . . . . . 103
3.19 Empirical regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.20 A simulated data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.21 A kernel smooth of the simulated data set. . . . . . . . . . . . . . . . . . . . 106