Kernel Smoothing Principles, Methods and Applications

Comprehensive theoretical overview of kernel smoothing methods with motivating examples

Kernel smoothing is a flexible nonparametric curve estimation method that is applicable when parametric descriptions of the data are not sufficiently adequate. This book explores theory and methods of kernel smoothing in a variety of contexts, considering independent and correlated data e.g. with short-memory and long-memory correlations, as well as non-Gaussian data that are transformations of latent Gaussian processes. These types of data occur in many fields of research, e.g. the natural and the environmental sciences, and others. Nonparametric density estimation, nonparametric and semiparametric regression, trend and surface estimation in particular for time series and spatial data and other topics such as rapid change points, robustness etc. are introduced alongside a study of their theoretical properties and optimality issues, such as consistency and bandwidth selection.

Addressing a variety of topics, Kernel Smoothing: Principles, Methods and Applications offers a user-friendly presentation of the mathematical content so that the reader can directly implement the formulas using any appropriate software. The overall aim of the book is to describe the methods and their theoretical backgrounds, while maintaining an analytically simple approach and including motivating examples—making it extremely useful in many sciences such as geophysics, climate research, forestry, ecology, and other natural and life sciences, as well as in finance, sociology, and engineering.

• A simple and analytical description of kernel smoothing methods in various contexts
• Presents the basics as well as new developments
• Includes simulated and real data examples

Kernel Smoothing: Principles, Methods and Applications is a textbook for senior undergraduate and graduate students in statistics, as well as a reference book for applied statisticians and advanced researchers. 

Sucharita Ghosh, PhD, is a statistician at the Swiss Federal Research Institute WSL, Switzerland. She also teaches graduate level Statistics in the Department of Mathematics, Swiss Federal Institute of Technology in Zurich. She obtained her doctorate in Statistics from the University of Toronto, Masters from the Indian Statistical Institute and B.Sc. from Presidency College, University of Calcutta, India. She was a Statistics faculty member at Cornell University and has held various short-term and long-term visiting faculty positions at universities such as the University of North Carolina at Chapel Hill and University of York, UK. She has also taught Statistics to undergraduate and graduate students at a number of universities, namely in Canada (Toronto), USA (Cornell, UNC Chapel Hill), UK (York), Germany (Konstanz) and Switzerland (ETH Zurich). Her research interests include smoothing, integral transforms, time series and spatial data analysis, having applications in a number of areas including the natural sciences, finance and medicine among others.

Preface ix

Density Estimation 1

1.1 Introduction 1

1.1.1 Orthogonal polynomials 2

1.2 Histograms 8

1.2.1 Properties of the histogram 9

1.2.2 Frequency polygons 14

1.2.3 Histogram bin widths 15

1.2.4 Average shifted histogram 19

1.3 Kernel density estimation 19

1.3.1 Naive density estimator 21

1.3.2 Parzen–Rosenblatt kernel density estimator 25

1.3.3 Bandwidth selection 43

1.4 Multivariate density estimation 53

Nonparametric Regression 59

2.1 Introduction 59

2.1.1 Method of least squares 60

2.1.2 Influential observations 70

2.1.3 Nonparametric regression estimators 71

2.2 Priestley–Chao regression estimator 73

2.2.1 Weak consistency 77

2.3 Local polynomials 80

2.3.1 Equivalent kernels 84

2.4 Nadaraya–Watson regression estimator 87

2.5 Bandwidth selection 93

2.6 Further remarks 99

2.6.1 Gasser–M¨uller estimator 99

2.6.2 Smoothing splines 100

2.6.3 Kernel efficiency 103

Trend Estimation 105

3.1 Time series replicates 105

3.1.1 Model 111

3.1.2 Estimation of common trend function 114

3.1.3 Asymptotic properties 114

3.2 Irregularly spaced observations 120

3.2.1 Model 122

3.2.2 Derivatives, distribution function, and quantiles 125

3.2.3 Asymptotic properties 129

3.2.4 Bandwidth selection 137

3.3 Rapid change points 141

3.3.1 Model and definition of rapid change 144

3.3.2 Estimation and asymptotics 145

3.4 Nonparametric M-estimation of a trend function 149

3.4.1 Kernel-based M-estimation 149

3.4.2 Local polynomial M-estimation 154

Semiparametric Regression 157

4.1 Partial linear models with constant slope 157

4.2 Partial linear models with time-varying slope 160

4.2.1 Estimation 165

4.2.2 Assumptions 166

4.2.3 Asymptotics 171

Surface Estimation 181

5.1 Introduction 181

5.2 Gaussian subordination 193

5.3 Spatial correlations 195

5.4 Estimation of the mean and consistency 197

5.4.1 Asymptotics 197

5.5 Variance estimation 203

5.6 Distribution function and spatial Gini index 206

5.6.1 Asymptotics 213

References 217

Author Index 243

Subject Index 251

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