Local Polynomial Modelling and Its Applications: Monographs on Statistics and Applied Probability 66

✍ Scribed by Jinqing Fan; Gijbels, Irene

• Publisher: CRC Press;Routledge
• Year: 2018
• Tongue: English
• Leaves: 358
• Series: Monographs on Statistics and Applied Probability, 66
• Category: Library

✦ Synopsis

Data-analytic approaches to regression problems, arising from many scientific disciplines are described in this text. The aim of these nonparametric methods is to relax assumptions on the form of a regression function, and to let data search for a suitable function that describes the data well. The use of these nonparametric functions with parametric techniques can yield very powerful data anlysis tools.

✦ Table of Contents

Content: Cover
Dedication
Title Page
Copyright Page
Table of Contents
Preface
1: Introduction
1.1 From linear regression to nonlinear regression
1.2 Local modelling
1.3 Bandwidth selection and model camplexity
1.4 Scope of the book
1.5 Implementation of nonparametric techniques
1.6 Purther reading
2: Overview of existing methods
2.1 Introduction
2.2 Kernel estimators
2.2.1 Nadaraya-Watson estimator
2.2.2 Gasser-Miiller estimator
2.2.3 Limitations of a local eonstant fit
2.3 Local polynomial fitting and derivative estimation
2.3.1 Local polynomial fitting 2.3.2 Derivative estimation2.4 Locally weighted scatter plot smoothing
2.4.1 Robust locally weighted regression
2.4.2 An example
2.5 Wavelet thresholding
2.5.1 Orthogonal series based methods
2.5.2 Basic ingredient of multiresolution analysis
2.5.3 Wavelet shrinkage estimator
2.5.4 Discrete wavelet transform
2.6 Spline smoothing
2.6.1 Polynomial spline
2.6.2 Smoothing spline
2.7 Density estimation
2.7.1 Kernel density estimation
2.7.2 Regression view of density estimation
2.7.3 Wavelet estimators
2.7.4 Logspline method
2.8 Bibliographie notes 3: Framework for local polynomial regression3.1 Introduetion
3.2 Advantages of loeal polynomial fitting
3.2.1 Bias and variance
3.2.2 Equivalent kernels
3.2.3 Ideal ehoice of bandwidth
3.2.4 Design adaptation property
3.2.5 Automatie boundary earpentry
3.2.6 Universal optimal weighting scheme
3.3 Whieh order of polynomial fit to use?
3.3.1 Inereases of variability
3.3.2 It is an odd world
3.3.3 Variable order approximation
3.4 Best linear smoothers
3.4.1 Best linear smoother at interior: optimal rates and eonstants
3.4.2 Best linear smoother at boundary 3.5 Minimax efficieney of local polynomial fitting3.5.1 Modulus of eontinuity
3.5.2 Best rates and nearly best Constant
3.6 Fast eomputing algorithms
3.6.1 Binning implementation
3.6.2 Updating algorithm
3.7 Complements
3.8 Bibliographie notes
4: Automatic determination of model complexity
4.1 Introduetion
4.2 Rule of thumb for bandwidth seleetion
4.3 Estimated bias and variance
4.4 Confidenee intervals
4.5 Residua! squares eriterion
4.5.1 Residua! squares eriterion
4.5.2 eonstant bandwidth seleetion
4.5.3 Variable bandwidth seleetion
4.5.4 Computation and related issues 4.6 Refined bandwidth selection4.6.1 Improving rates of convergence
4.6.2 eonstant bandwidth selection
4.6.3 Variable bandwidth selection
4.7 Variable bandwidth and spatial adaptation
4.7.1 Qualification of spatial adaptation
4.7.2 Comparison with wavelets
4.8 Smoothing techniques in use
4.8.1 Example 1: modelling and model diagnostics
4.8.2 Example 2: comparing two treatments
4.8.3 Example 3: analyzing a longitudinal data set
4.9 A blueprint for local modelling
4.10 Other existing methods
4.10.1 Normal reference method
4.10.2 Cross-validation
4.10.3 Nearest neighbor bandwidth