Table Of Content

Statistics for Industry and Technology Series Editor N. Balakrishnan McMaster University Department of Mathematics and Statistics 1280 Main Street West Hamilton, Ontario L8S 4K1 Canada Editorial Advisory Board Max Engelhardt EG&G Idaho, Inc. Idaho Falls, ID 83415 Harry F. Martz Group A-1 MS F600 Los Alamos National Laboratory Los Alamos, NM 87545 Gary C. McDonald NAO Research & Development Center 30500 Mound Road Box 9055 Warren, MI 48090-9055 Peter R. Nelson Department of Mathematcal Sciences Clemson University Martin Hall Box 341907 Clemson, SC 29634-1907 Kazuyuki Suzuki Communication & Systems Engineering Department University of Electro Communications 1-5-1 Chofugaoka Chofu-shi Tokyo 182 Japan Goodness-of-Fit Tests and Model Validity C. Huber-Carol N. Balakrishnan M.S. Nikulin M. Mesbah Editors Springer Science+Business Media, LLC C. Huber-Carol N. Balakrishnan Laboratoire de Statistique Medicale Department of Mathematics and Statistics Universite Rene Descartes—Paris 5 McMaster University 75006 Paris Hamilton, Ontario L8S 4K1 France Canada M. S. Nikulin M. Mesbah Laboratoire de Statistique Appliquee Laboratoire Statistique Mathematique Universite de Bretagne Sud Universite Bordeaux 2 56 000 Vannes 33076 Bordeaux Cedex France France and Laboratory of Statistical Methods V. Steklov Mathematical Institute 191011 St. Petersburg Russia Library of Congress Cataloging-in-Publication Data A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA. AMS Subject Classifications: 62-06, 62F03 Printed on acid-free paper. ÜL5) ® ©2002 Springer Science+Business Media New York U^f) Originally published by Birkhäuser Boston in 2002 Softcover reprint of the hardcover 1st edition 2002 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar method ology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. ISBN 978-1-4612-6613-6 ISBN 978-1-4612-0103-8 (eBook) DOI 10.1007/978-1-4612-0103-8 Typeset by the editors in IATEX. 9 8 7 6 5 4 3 2 1 Contents Preface xvii Contributors xix List of Tables xxvii List of Figures xxxiii PART I: HISTORY AND FUNDAMENTALS 1 Karl Pearson and the Chi-Squared Test 3 D. R. Cox 1.1 Karl Pearson 1857-1937: Background to the Chi-Squared Paper 3 1.2 K. P.: After Chi-Squared 5 1.3 The 1900 Paper 5 1.4 Importance of the Chi-Squared Test 6 References 8 2 Karl Pearson Chi-Square Test-The Dawn of Statistical Inference 9 C. R. Rao 2.1 Introduction 9 2.2 Large Sample Criteria: The Holy 'frinity 11 2.2.1 Likelihood ratio criterion 11 2.2.2 Wald test 12 2.2.3 Rao's score test 12 2.3 Specification Tests for a Multinomial Distribution 13 2.3.1 Test of a simple hypothesis 13 2.3.2 Tests of a composite hypothesis 14 2.3.3 Test for goodness-of-fit in a subset of cells 15 2.3.4 Analysis of chi-square 17 2.3.5 Some applications of the chi-square test 18 2.4 Other Tests of Goodness-of-Fit 18 v vi Contents 2.5 Specification Tests for Continuous Distributions 20 References 22 3 Approximate Models 25 Peter J. Huber 3.1 Models 25 3.2 Bayesian Modeling 27 3.3 Mathematical Statistics and Approximate Models 29 3.4 Statistical Significance and Relevance 31 3.5 Composite Models 32 3.6 The Role of Simulation 38 3.7 Summary Conclusions 40 References 40 PART II: CHI-SQUARED TEST 4 Partitioning the Pearson-Fisher Chi-Squared Goodness-of-Fit Statistic 45 G. D. Rayner 4.1 Introduction 45 4.2 Neyman Smooth Goodness-of-Fit Tests 46 4.2.1 Smooth goodness-of-fit tests for categorized data 47 4.2.2 Partitioning the Pearson-Fisher chi-squared statistic 48 4.3 Constructing the Pearson-Fisher Decomposition 49 4.4 Simulation Study 50 4.5 Results and Discussion 51 References 55 5 Statistical Tests for Normal Family in Presence of Outlying Observations 57 A i"cha Z erbet 5.1 The Chi-Squared Test of Normality in the Univariate Case 57 5.1.1 Example: Analysis of the data of Milliken 59 5.2 Bol'shev Test for Outliers 59 5.2.1 Stages of applications of the test of Bol'shev 60 5.2.2 Example 2: Analysis of the data of Daniel (1959) 60 5.3 Power of the Chi-Squared Test 61 References 63 Contents Vll 6 Chi-Squared Test for the Law of Annual Death Rates: Case with Censure for Life Insurance Files 65 Leo Gerville-Reache 6.1 Introduction 65 6.2 Chi-Squared Goodness-of-Fit Test 66 6.2.1 Statistics with censure 66 6.2.2 Goodness-of-fit test for a composite hypothesis 67 6.3 Demonstration 68 References 69 PART III: GOODNESS-OF-FIT TESTS FOR PARAMETRIC DISTRIBUTIONS 7 Shapiro-Wilk Type Goodness-of-Fit Tests for Normality: Asymptotics Revisited 73 Pranab Kumar Sen 7.1 Introduction 73 7.2 Preliminary Notion 74 7.3 SOADR Results for BLUE and LSE 77 7.4 Asymptotics for W~ 81 7.5 Asymptotics Under Alternatives 85 References 87 8 A Test for Exponentiality Based on Spacings for Progressively Type-II Censored Data 89 N. Balakrishnan, H. K. T. Ng, and N. Kannan 8.1 Introduction 89 8.2 Progressive Censoring 91 8.3 Test for Exponentiality 92 8.3.1 Null distribution of T 93 8.4 Power Function Approximation and Simulation Results 95 8.4.1 Approximation of power function 95 8.4.2 Monte Carlo power comparison 97 8.5 Modified EDF and Shapiro-Wilk Statistics 98 8.6 Two-Parameter Exponential Case 99 8.7 Illustrative Examples 100 8.7.1 Example 1: One-parameter exponential case 100 8.7.2 Example 2: Two-parameter exponential case 101 8.8 Multi-Sample Extension 102 8.9 Conclusions 103 References 103 viii Contents 9 Goodness-of-Fit Statistics for the Exponential Distribution When the Data are Grouped 113 Sneh Gulati and Jordan Neus 9.1 Introduction 113 9.2 The Model and the Test Statistics 115 9.3 Asymptotic Distribution 116 9.4 Power Studies 119 References 122 10 Characterization Theorems and Goodness-of-Fit Tests 125 Carol E. Marchetti and Govind S. Mudholkar 10.1 Introduction and Summary 126 10.2 Characterization Theorems 127 10.2.1 Entropy characterizations 127 10.2.2 Statistical independence 128 10.3 Maximum Entropy Tests 130 10.4 Four Z Tests 131 10.5 Byproducts: The G-IG Analogies 134 References 137 11 Goodness-of-Fit Tests Based on Record Data and Generalized Ranked Set Data 143 Barry C. Arnold, Robert J. Beaver, Enrique Castillo, and Jose Maria Sarabia 11.1 Introduction 143 11.2 Record Data 144 11.3 Generalized Ranked Set Data 144 11.4 Power 150 11.5 Composite Null Hypotheses 154 11.6 Remarks 156 References 156 PART IV: REGRESSION AND GOODNESS-OF-FIT TESTS 12 Gibbs Regression and a Test of Goodness-of-Fit 161 Lynne Seymour 12.1 Introduction 161 12.2 The Motivation and the Model 162 12.3 Application and Evaluation of the Model 165 12.4 Discussion 169 References 170 13 A CLT for the L_2 Norm of the Regression Estimators Under a-Mixing: Application to G-O-F Tests 173 Cheikh A. T. Diack 13.1 Introduction 173 13.2 Estimators 174 13.3 A Limit Theorem 175 13.4 Inference 177 13.5 Proofs 178 References 183 14 Testing the Goodness-of-Fit of a Linear Model in N onparametric Regression 185 Zaher Mohdeb and Abdelkader Mokkadem 14.1 Introduction 185 14.2 The Test Statistic 186 14.3 Simulations 189 References 193 15 A New Test of Linear Hypothesis in Regression 195 Y. Baraud, S. Huet, and B. Laurent 15.1 Introduction 195 15.2 The Testing Procedure 196 15.2.1 Description of the procedure 197 15.2.2 Behavior of the test under the null hypothesis 198 15.2.3 A toy framework: The case of a known variance 198 15.3 The Power of the Test 198 15.3.1 The main result 198 15.3.2 Rates of testing 199 15.4 Simulations 201 15.4.1 The simulation experiment 201 15.4.2 The testing procedure 202 15.4.3 The test proposed by Horowitz and Spokoiny (2000) 202 15.4.4 Results of the simulation study 203 15.5 Proofs 203 15.5.1 Proof of Theorem 15.3.1 203 15.5.2 Proof of Corollary 15.3.1 204 References 206 x Contents PART V: GOODNESS-OF-FIT TESTS IN SURVIVAL ANALYSIS AND RELABILITY 16 Inference in Extensions of the Cox Model for Heterogeneous Populations 211 Odile Pons 16.1 Introduction 211 16.2 Non-Stationary Cox Model 212 16.3 Varying-Coefficient Cox Model 219 References 224 17 Assumptions of a Latent Survival Model 227 Mei-Ling Ting Lee and G. A. Whitmore 17.1 Introduction 227 17.2 Latent Survival Model 228 17.3 Data and Parameter Estimation 229 17.4 Model Validation Methods 230 17.5 Remedies to Achieve a Better Model Fit 233 References 235 18 Goodness-of-Fit Testing for the Cox Proportional Hazards Model 237 K arthik Devarajan and Nader Ebrahimi 18.1 Introduction 237 18.2 Goodness-of-Fit Testing for the Cox PH Model 240 18.3 Comparison of the Proposed Goodness-of-Fit Test with Existing Methods 242 18.4 Illustration of the Goodness-of-Fit Test using Real-Life Data 249 18.5 Concluding Remarks 250 References 251 19 A New Family of Multivariate Distributions for Survival Data 255 Shulamith T. Gross and Catherine Huber-Carol 19.1 Introduction 255 19.2 Frailty Models: An Overview 255 19.3 The Model 257 19.4 An Application to Skin Grafts Rejection 261 19.4.1 Description of the data 261 References 264