Kernel Smoothing

224 Pages·1995·6.822 MB·English

by  M. P. Wand, M. C. Jones (auth.)

Kernel Smoothing
MONOGRAPHS ON
STATISTICS AND APPLIED PROBABILITY
General Editors
D.R. Cox, D.V. Hinkley, N. Reid, D.B. Rubin and B.W. Silvennan
1 Stochastic Population Models in Ecology and Epidemiology
M.S. Bartlett (1960)
2 Queues D.R. Cox and W.L. Smith (1961)
3 Monte Carlo Methods J M. Hammersley and D.C. Handscomb (1964)
4 The Statistical Analysis of Series of Events D.R. Cox and P.A.W. Lewis
(1966)
5 Population Genetics W.J. Ewens (1969)
6 Probability, Statistics and Time M.S. Bartlett (1975)
7 Statistical Inference S.D. Silvey (1975)
8 The Analysis of Contingency Tables B.S. Everitt (1977)
9 Multivariate Analysis in Behavioural ResearchA.E. Maxwell (1977)
10 Stochastic Abundance Models S. Engen (1978)
11 Some Basic Theory for Statistical Inference E.J.G. Pitman (1978)
12 Point Processes D.R. Cox and V. Isham (1980)
13 Identification of Outliers D.M. Hawkins (1980)
14 Optimal Design S.D. Silvey (1980)
15 Finite Mixture Distributions B.S. Everitt and D.J. Hand (1981)
16 ClassificationA.D. Gordon (1981)
17 Distribution-free Statistical Methods J.S. Maritz (1981)
18 Residuals and Influence in Regression R.D. Cook and S. Weisberg
(1982)
19 Applications of Queueing Theory G. F. Newell (1982)
20 Risk Theory, 3rd edition R.E. Beard, T. Pentikiiinen and E. Pesonen
(1984)
21 Analysis of Survival Data D.R. Cox and D. Oakes (1984)
22 An Introduction to Latent Variable Models B.S. Everitt (1984)
23 Bandit Problems D.A. Berry and B. Fristedt (1985)
24 Stochastic Modelling and Control M.H.A. Davis and R. Vinter (1985)
25 The Statistical Analysis of Compositional Datal. Aitchison (1986)
26 Density Estimation for Statistical and Data Analysis
B.W. Silverman (1986)
27 Regression Analysis with Application G.B. Wetherill (1986)
28 Sequential Methods in Statistics, 3rd edition G.B. Wetherill (1986)
29 Tensor Methods in Statistics P. McCullagh (1987)
30 Transformation and Weighting in Regression R.I. Carroll
and D. Ruppert (1988)
31 Asymptotic Techniques for Use in Statistics O.E. Bamdorff-Nielsen
and D.R. Cox (1989)
32 Analysis of Binary Data, 2nd edition D.R. Cox and E.J. Snell (1989)
33 Analysis of Infectious Disease Data N.G. Becker (1989)
34 Design and Analysis of Cross-Over Trials B. Jones and M.G. Kenwara
(1989)
35 Empirical Bayes Method, 2nd edition J.S. Maritz and T. Lwin (1989)
36 Symmetric Multivariate and Related Distributions K.-T. Fang,
S. Kotz and K. Ng (1989)
37 Generalized Linear Models, 2nd edition P. McCullagh
and J.A. Neider (1989)
38 Cyclic DesignsJ.A. John (1987)
39 Analog Estimation Methods in Econometrics C.F. Manski (1988)
40 Subset Selection in Regression A.!. Miller (1990)
41 Analysis of Repeated Measures M. Crowder and D.J.Hand (1990)
42 Statistical Reasoning with Imprecise Probabilities P. Walley (1990)
43 Generalized Additive Models T.J. Hastie and R.I. Tibshirani (1990)
44 Inspection Errors for Attributes in Quality Control N.L. Johnson,
S.KotzandX. Wu (1991)
45 The Analysis of Contingency Tables, 2nd edition B.S. Everitt (1992)
46 The Analysis of Quantal Response Data B.J.T. Morgan (1992)
47 Longitudinal Data with Serial Correlation: A State-space Approach
R.H. Jones (1993)
48 Differential Geometry and Statistics M.K. Murray and J. W. Rice (1993)
49 Markov Models and Optimization M.H.A. Davis
50 Networks and Chaos-Statistical and Probabilistic Aspects
O.E. Bamdorff-Nielsen, J.L. Jensen and W.S. Kendall (1993)
51 Number Theoretic Methods in Statistics K.-T. Fang andY. Wang
(1993)
52 Inference and Asymptotics O.E. Bamdorff-Nielsen and D.R. Cox
(1994)
53 Practical Risk Theory for Actuaries C.D. Daykin, T. Pentikiiinen
and M. Pesonen (1993)
54 Statistical Concepts and Applications in Medicine J. Aitchison
and I.J. Lauder (1994)
55 Predictive InferenceS. Geisser (1993)
56 Model Free Curve Estimation M. Tarter and M. Lock (1994)
57 An Introduction to the Bootstrap B. Efron and R. Tibshirani (1993)
58 Nonparametric Regression and Generalized Linear Models
P.J. Green and B.W. Silverman (1994)
59 Multidimensional Scaling T.F. Cox and M.A.A. Cox (1995)
60 Kernel SmoothingM.P. Wand andM.C.Jones (1995)
(Full details concerning this series are available from the Publishers.)
Kernel Smoothing
M.P. WAND
Australian Graduate School ofM anagement
University ofN ew South Wales
Australia
M.C.JONES
Department of Statistics
The Open University
Milton Keynes, UK
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
First edition 1995
©M.P. Wand and M. C. Jones 1995
Originally published by Chapman & Hall, New York in 1995
Softcover reprint of the hardcover I st edition 1995
ISBN 978-0-412-55270-0 ISBN 978-1-4899-4493-1 (eBook)
DOI 10.1007/978-1-4899-4493-1
Apart from any fair dealing for the purposes of research or private study, or criticism or
review, as permitted under the UK Copyright Designs and Patents Act, 1988, this
publication may not be reproduced, stored, or transmitted, in any form or by any means,
without the prior permission in writing of the publishers, or in the case of reprographic
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appropriate Reproduction Rights Organization outside the UK. Enquiries concerning
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address printed on this page.
The publisher makes no representation, express or implied, with regard to the accuracy of
the information contained in the book and cannot accept any legal responsibility or liability
for any errors or omissions that may be made.
A catalogue record for this book is available from the British Library
@J Printed on permanent acid-free paper, manufactured in accordance with ANSIINISO
Z39.48-1992 and ANSI/NISO Z39.48-1984 (Permanence of Paper).
Contents
Preface xi
1 Introduction 1
1.1 Introduction 1
1.2 Density estimation and histograms 5
1.3 About this book 7
1.4 Options for reading this book 9
1.5 Bibliographical notes 9
2 Univariate kernel density estimation 10
2.1 Introduction 10
2.2 The univariate kernel density estimator 11
2.3 The MSE and MISE criteria 14
2.4 Order and asymptotic notation; Taylor expansion 17
2.4.1 Order and asymptotic notation 17
2.4.2 Taylor expansion 19
2.5 Asymptotic MSE and MISE approximations 19
2.6 Exact MISE calculations 24
2.7 Canonical kernels and optimal kernel theory 28
2.8 Higher-order kernels 32
2.9 Measuring how difficult a density is to estimate 36
2.10 Modifications of the kernel density estimator 40
2.10.1 Local kernel density estimators 40
2.10.2 Variable kernel density estimators 42
2.10.3 Transformation kernel density estimators 43
2.11 Density estimation at boundaries 46
2.12 Density derivative estimation 49
2.13 Bibliographical notes 50
2.14 Exercises 52
3 Bandwidth selection 58
3.1 Introduction 58
3.2 Quick and simple bandwidth selectors 59
3.2.1 Nor mal scale rules 60
vii
viii CONTENTS
3.2.2. Oversmoothed bandwidth selection rules 61
3.3 Least squares cross-validation 63
3.4 Biased cross-validation 65
3.5 Estimation of density functionals 67
3.6 Plug-in bandwidth selection 71
3.6.1 Direct plug-in rules 71
3.6.2 Solve-the-equation rules 74
3. 7 Smoothed cross-validation bandwidth selection 75
3.8 Comparison of bandwidth selectors 79
3.8.1 Theoretical performance 79
3.8.2 Practical advice 85
3.9 Bibliographical notes 86
3.10 Exercises 88
4 Multivariate kernel density estimation 90
4.1 Introduction 90
4.2 The multivariate kernel density estimator 91
4.3 Asymptotic MISE approximations 94
4.4 Exact MISE calculations 101
4.5 Choice of a multivariate kernel 103
4.6 Choice of smoothing parametrisation 105
4. 7 Bandwidth selection 108
4.8 Bibliographical notes 110
4.9 Exercises 110
5 Kernel regression 114
5.1 Introduction 114
5.2 Local polynomial kernel estimators 116
5.3 Asymptotic MSE approximations: linear case 120
5.3.1 Fixed equally spaced design 120
5.3.2 Random design 123
5.4 Asymptotic MSE approximations: general case 125
5.5 Behaviour near the boundary 126
5.6 Comparison with other kernel estimators 130
5.6.1 Asymptotic comparison 130
5.6.2 Effective kernels 133
5. 7 Derivative estimation 135
5.8 Bandwidth selection 138
5.9 Multivariate nonparametric regression 140
5.10 Bibliographical notes 141
5.11 Exercises 143
6 Selected extra topics 146
6.1 Introduction 146
6.2 Kernel density estimation in other settings 147
CONTENTS ix
6.2.1 Dependent data 147
6.2.2 Length biased data 150
6.2.3 Right-censored data 154
6.2.4 Data measured with error 156
6.3 Hazard function estimation 160
6.4 Spectral density estimation 162
6.5 Likelihood-based regression models 164
6.6 Intensity function estimation 167
6. 7 Bibliographical notes 169
6.8 Exercises 170
Appendices 172
A Notation 172
B Tables 175
C Facts about normal densities 177
C.1 Univariate normal densities 177
C.2 Multivariate normal densities 180
C.3 Bibliographical notes 181
D Computation of kernel estimators 182
D.1 Introduction 182
D.2 The binned kernel density estimator 183
D.3 Computation of kernel functional estimates 188
D.4 Computation of kernel regression estimates 189
D.5 Extension to multivariate kernel smoothing 191
D.6 Computing practicalities 192
D.7 Bibliographical notes 192
References 193
Index 208
TO
CHRISTINE, PAUL AND HANDAN
BETTY, FRANK AND PING