Smoothing Techniques: With Implementation in S

ISBN-13: 9781461287681 / Angielski / Miękka / 2011 / 262 str.

Wolfgang H. Rdle

Opis

The author has attempted to present a book that provides a non-technical introduction into the area of non-parametric density and regression function estimation. The application of these methods is discussed in terms of the S computing environment. Smoothing in high dimensions faces the problem of data sparseness. A principal feature of smoothing, the averaging of data points in a prescribed neighborhood, is not really practicable in dimensions greater than three if we have just one hundred data points. Additive models provide a way out of this dilemma; but, for their interactiveness and recursiveness, they require highly effective algorithms. For this purpose, the method of WARPing (Weighted Averaging using Rounded Points) is described in great detail.

Spis treści

I. Density Smoothing.- 1. The Histogram.- 1.0 Introduction.- 1.1 Definitions of the Histogram.- The Histogram as a Frequency Counting Curve.- The Histogram as a Maximum Likelihood Estimate.- Varying the Binwidth.- 1.2 Statistics of the Histogram.- 1.3 The Histogram in S.- 1.4 Smoothing the Histogram by WARPing.- WARPing Algorithm.- WARPing in S.- Exercises.- 2. Kernel Density Estimation.- 2.0 Introduction.- 2.1 Definition of the Kernel Estimate.- Varying the Kernel.- Varying the Bandwidth.- 2.2 Kernel Density Estimation in S.- Direct Algorithm.- Implementation in S.- 2.3 Statistics of the Kernel Density.- Speed of Convergence.- Confidence Intervals and Confidence Bands.- 2.4 Approximating Kernel Estimates by WARPing.- 2.5 Comparison of Computational Costs.- 2.6 Comparison of Smoothers Between Laboratories.- Keeping the Kernel Bias the Same.- Keeping the Support of the Kernel the Same.- Canonical Kernels.- 2.7 Optimizing the Kernel Density.- 2.8 Kernels of Higher Order.- 2.9 Multivariate Kernel Density Estimation.- Same Bandwidth in Each Component.- Nonequal Bandwidths in Each Component.- A Matrix of Bandwidths.- Exercises.- 3. Further Density Estimators.- 3.0 Introduction.- 3.1 Orthogonal Series Estimators.- 3.2 Maximum Penalized Likelihood Estimators.- Exercises.- 4. Bandwidth Selection in Practice.- 4.0 Introduction.- 4.1 Kernel Estimation Using Reference Distributions.- 4.2 Plug-In Methods.- 4.3 Cross-Validation.- 4.3.1 Maximum Likelihood Cross-Validation.- Direct Algorithm.- 4.3.2 Least-Squares Cross-Validation.- Direct Algorithm.- 4.3.3 Biased Cross-Validation.- Algorithm.- 4.4 Cross-Validation for WARPing Density Estimation.- 4.4.1 Maximum Likelihood Cross-Validation.- 4.4.2 Least-Squares Cross-Validation.- Algorithm.- Implementation in S.- 4.4.3 Biased Cross-Validation.- Algorithm.- Implementation in S.- Exercises.- II. Regression Smoothing.- 5. Nonparametric Regression.- 5.0 Introduction.- 5.1 Kernel Regression Smoothing.- 5.1.1 The Nadaraya-Watson Estimator.- Direct Algorithm.- Implementation in S.- 5.1.2 Statistics of the Nadaraya-Watson Estimator.- 5.1.3 Confidence Intervals.- 5.1.4 Fixed Design Model.- 5.1.5 The WARPing Approximation.- Basic Algorithm.- Implementation in S.- 5.2 k-Nearest Neighbor (k-NN).- 5.2.1 Definition of the k-NN Estimate.- 5.2.2 Statistics of the k-NN Estimate.- 5.3 Spline Smoothing.- Exercises.- 6. Bandwidth Selection.- 6.0 Introduction.- 6.1 Estimates of the Averaged Squared Error.- 6.1.0 Introduction.- 6.1.1 Penalizing Functions.- 6.1.2 Cross-Validation.- Direct Algorithm.- 6.2 Bandwidth Selection with WARPing.- Penalizing Functions.- Cross-Validation.- Basic Algorithm.- Implementation in S.- Applications.- Exercises.- 7. Simultaneous Error Bars.- 7.1 Golden Section Bootstrap.- Algorithm for Golden Section Bootstrapping.- Implementation in S.- 7.2 Construction of Confidence Intervals.- Exercises.- Tables.- Solutions.- List of Used S Commands.- Symbols and Notation.- References.

O autorze

The book is addressed to both the theoretical and applied statistician. It can be used as an undergraduate text.