Statistical modelling with quantile functions

317 Pages·2000·2.276 MB·English

by  Warren Gilchrist

Statistical
Modelling
Quantile
with
Functions
© 2000 by Chapman & Hall/CRC
Statistical
Modelling
Quantile
with
Functions
Warren G. Gilchrist
Emeritus Professor
Sheffield Hallam University
United Kingdom
CHAPMAN & HALL/CRC
Boca Raton London New York Washington, D.C.
© 2000 by Chapman & Hall/CRC
Library of Congress Cataloging-in-Publication Data
Gilchrist, Warren, 1932-
Statistical modelling with quantile functions / Warren G. Gilchrist.
p. cm.
Includes bibliographical references and index.
ISBN 1-58488-174-7 (alk. paper)
1. Distribution (Probability theory) 2. Sampling (Statistics) I. Title.
QA276.7 .G55 2000
519.2—dc21 00-023728
CIP
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© 2000 by Chapman & Hall/CRC
No claim to original U.S. Government works
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© 2000 by Chapman & Hall/CRC
Contents
List of Figures xi
List of Tables xv
Preface xix
1 An Overview 1
1.1 Introduction 1
1.2 The data and the model 3
1.3 Sample properties 3
1.4 Modelling the population 9
The cumulative distribution function 9
The probability density function 11
The quantile function 12
The quantile density function 14
1.5 A modelling kit for distributions 15
1.6 Modelling with quantile functions 17
1.7 Simple properties of population quantile functions 24
1.8 Elementary model components 28
1.9 Choosing a model 31
1.10 Fitting a model 34
1.11 Validating a model 39
1.12 Applications 39
1.13 Conclusions 41
2 Describing a Sample 43
2.1 Introduction 43
2.2 Quantiles and moments 44
2.3 The five-number summary and measures of spread 50
2.4 Measures of skewness 53
2.5 Other measures of shape 55
© 2000 by Chapman & Hall/CRC
vi
2.6 Bibliographic notes 57
2.7 Problems 59
3 Describing a Population 61
3.1 Defining the population 61
3.2 Rules for distributional model building 62
The reflection rule 62
The addition rule 63
The multiplication rule for positive variables 63
The intermediate rule 63
The standardization rule 64
The reciprocal rule 65
The Q-transformation rule 65
The uniform transformation rule 66
The p-transformation rule 66
3.3 Density functions 67
The addition rule for quantile density functions 67
3.4 Population moments 68
3.5 Quantile measures of distributional form 71
3.6 Linear moments 74
L-moments 74
Probability-weighted moments 77
3.7 Problems 79
4 Statistical Foundations 83
4.1 The process of statistical modelling 83
4.2 Order statistics 84
The order statistics distribution rule 86
The median rankit rule 89
4.3 Transformation 90
The median transformation rule 94
4.4 Simulation 94
4.5 Approximation 97
4.6 Correlation 100
4.7 Tailweight 102
Using tail quantiles 103
The TW(p) function 103
Limiting distributions 105
4.8 Quantile models and generating models 106
4.9 Smoothing 108
4.10 Evaluating linear moments 111
4.11 Problems 113
© 2000 by Chapman & Hall/CRC
vii
5 Foundation Distributions 117
5.1 Introduction 117
5.2 The uniform distribution 117
5.3 The reciprocal uniform distribution 118
5.4 The exponential distribution 119
5.5 The power distribution 120
5.6 The Pareto distribution 121
5.7 The Weibull distribution 122
5.8 The extreme, type 1, distribution and the Cauchy
distribution 122
5.9 The sine distribution 124
5.10 The normal and log-normal distributions 125
5.11 Problems 128
6 Distributional Model Building 131
6.1 Introduction 131
6.2 Position and scale change — generalizing 131
6.3 Using addition — linear and semi-linear models 133
6.4 Using multiplication 140
6.5 Using Q-transformations 141
6.6 Using p-transformations 143
6.7 Distributions of largest and smallest observations 145
6.8 Conditionally modified models 147
Conditional probabilities 147
Blipped distributions 148
Truncated distributions 148
Censored data 150
6.9 Conceptual model building 150
6.10 Problems 152
7 Further Distributions 155
7.1 Introduction 155
7.2 The logistic distributions 155
7.3 The lambda distributions 156
The three-parameter, symmetric, Tukey-lambda
distribution 157
The four-parameter lambda 158
The generalized lambda 160
The five-parameter lambda 163
7.4 Extreme value distributions 164
7.5 The Burr family of distributions 167
© 2000 by Chapman & Hall/CRC
viii
7.6 Sampling distributions 168
7.7 Discrete distributions 169
Introduction 169
The geometric distribution 170
The binomial distribution 171
7.8 Problems 172
8 Identification 173
8.1 Introduction 173
8.2 Exploring the data 173
The context 173
Numerical summaries 174
General shape 175
Skewness 175
Tail shape 176
Interpretation 176
8.3 Selecting the models 177
Starting points 177
Identification plots 178
8.4 Identification involving component models 184
8.5 Sequential model building 186
8.6 Problems 190
9 Estimation 193
9.1 Introduction 193
9.2 Matching methods 193
9.3 Methods based on lack of fit criteria 198
9.4 The method of maximum likelihood 207
9.5 Discounted estimation 210
9.6 Intervals and regions 213
9.7 Initial estimates 217
9.8 Problems 218
10 Validation 223
10.1 Introduction 223
10.2 Visual validation 224
Q-Q plots 224
Density probability plots 224
Residual plots 226
Further plots 227
Unit exponential spacing control chart 227
© 2000 by Chapman & Hall/CRC
ix
10.3 Application validation 228
10.4 Numerical supplements to visual validation 230
10.5 Testing the model 230
Goodness-of-fit tests 231
Testing using the uniform distribution 231
Tests based on confidence intervals 232
Tests based on the criteria of fit 232
10.6 Problems 235
11 Applications 237
11.1 Introduction 237
11.2 Reliability 237
Definitions 237
p-Hazards 238
11.3 Hydrology 241
11.4 Statistical process control 243
Introduction 243
Capability 243
Control charts 245
11.5 Problems 247
12 Regression Quantile Models 251
12.1 Approaches to regression modelling 251
12.2 Quantile autoregression models 260
12.3 Semi-linear and non-linear regression quantile
functions 261
12.4 Problems 266
13 Bivariate Quantile Distributions 269
13.1 Introduction 269
13.2 Polar co-ordinate models 271
The circular distributions 271
The Weibull circular distribution 274
The generalized Pareto circular distribution 275
The elliptical family of distributions 277
13.3 Additive models 279
13.4 Marginal/conditional models 280
13.5 Estimation 281
13.6 Problems 285
14 A Postscript 287
© 2000 by Chapman & Hall/CRC
x
Appendix 1 Some Useful Mathematical Results 293
Definitions 293
Series 294
Definite Integrals 294
Indefinite Integrals 294
Appendix 2 Further Studies in the Method
of Maximum Likelihood 295
Appendix 3 Bivariate Transformations 299
References 301
© 2000 by Chapman & Hall/CRC
List of Figures
(a) Flood data — x against p; (b) Flood data — p against x 5
(a) Flood data — Dp/Dx against mid-x (b) Flood data — Dx/Dp against
mid-p 7
Flood data — smoothed Dp/Dx against mid-p 8
A cumulative distribution function, F(x) 10
A probability density function, f(x) 12
A quantile function, Q(p) 13
PDF of the reflected exponential 18
(a) Quantile functions of the exponential and reflected exponential; (b)
Addition of exponential and reflected exponential
quantile functions 19
Addition of quantile density functions 20
The logistic distribution 20
The uniform and logistic distribution 21
(a) The power, Pareto and power × Pareto distribution quantile
functions. (b) The PDF for the power–Pareto
distribution 23
The p-PDF for the skew logistic distribution 27
p-PDFs for some basic models 30
Flood data — Fit-observation plot for a Weibull distribution 32
Flood data — (a) Observation-fit plots for two
models. (b) Quantile density plots for two models
and the data 33
© 2000 by Chapman & Hall/CRC