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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 This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without prior permission in writing from the publisher. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe. © 2000 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-174-7 Library of Congress Card Number 00-023728 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0 Printed on acid-free paper © 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