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Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Distribution Theory for Tests Based on the Sample Distribution Function Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php J. DURBIN London School of Economics and Politica) Science University of London Distribution Theory for Tests Based on the Sample Distribution Function ^iam SOCIETY FOR INDUSTRIAL AND APPLIED MATHEMATICS PHILADELPHIA Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS A series of lectures on topics of current research interest in applied mathematics under the direction of the Conference Board of the Mathematical Sciences, supported by the National Science Foundation and published by SIAM. The Numerical Solution of Elliptic Equations D. V. LINDLEY, Bayesian Statistics, A Review R. S. VARGA, Functional Analysis and Approximation Theory in Numerical Analysis R. R. BAHADUR, Some Limit Theorems in Statistics PATRICK BILLINGSLEY, Weak Convergence of Measures: Applications in Probability J. L. LIONS, Sorne Aspects of the Optietal Control of Distributed Parameter Systems ROGER PENROSE, Techniques ofDifferential Topology in Relativity HERMAN CHERNOFF, Sequential Analysis and Optima! Design J. DURBIN, Distribution Theory for Tests Based on the Sample Distribution Function SOL I. RUBINOW, Mathematical Problems in the Biological Sciences P. D. LAX, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves I. J. SCHOENBERG, Cardinal Spline Interpolation IVAN SINGER, The Theory of Best Approximation and Functional Analysis WERNER C. RHEINBOLDT, Methods of Solving Systems of Nonlinear Equations HANS F. WEINBERGER, Variational Methods for Eigenvalue Approximation R. TYRRELL ROCKAFELLAR, Conjugate Duality and Optimization SIR JAMES LIGHTHILL, Mathematical Biofluiddynamics GERARD SALTON, Theory of Indexing CATHLEEN S. MORAWETZ, Notes on Time Decay and Scattering for Some Hyperbolic Problems F. HOPPENSTEADT, Mathematical Theories of Populations: Demographics, Genetics and Epidemics RICHARD ASKEY, Orthogonal Polynomials and Special Functions L. E. PAYNE, Improperly Posed Problems in Partial Differential Equations S. ROSEN, Lectures on the Measurement and Evaluation of the Performance of Computing Systems HERBERT B. KELLER, Numerical Solution of Two Point Boundary Value Problems J. P. LASALLE, The Stability of Dynamical Systems D. GOTTLIEB AND S. A. ORSZAG, Numerical Analysis of Spectral Methods: Theory and Applications PETER J. HUBER, Robust Statistical Procedures HERBERT SOLOMON, Geometric Probability FRED S. ROBERTS, Graph Theory and Its Applications to Problems of Society JURIS HARTMANIE, Feasible Computations and Provable Complexity Properties ZOHAR MANNA, Lectures on the Logic of Computer Programming ELLIS L. JOHNSON, Integer Programming: Facets, Subadditivity, and Duality for Group and Semi-Group Problems SHMUEL WINOGRAD, Arithmetic Complexity of Computations J. F. C. KINGMAN, Mathematics of Genetic Diversity MORTON E. GURTIN, Topics in Finite Elasticity THOMAS G. KURTZ, Approximation of Population Processes JERROLD E. MARSDEN, Lectures on Geometric Methods in Mathematical Physics BRADLEY EFRON, The Jackknife, the Bootstrap, and Other Resampling Plans M. WOODROOFE, Nonlinear Renewal Theory in Sequential Analysis D. H. SATTINGER, Brand/zing in the Presence of Symmetry R. TEMAM, Navier—Stokes Equations and Nonlinear Functional Analysis GARRETT BIRKHOFF, Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php MIKLÓS CSÖRGO, Quantile Processes with Statistical Applications J. D. BUCKMASTER AND G. S. S. LUDFORD, Lectures on Mathematical Combustion R. E. TARJAN, Data Structures and Network Algorithms PAUL WALTMAN, Competition Models in Population Biology S. R. S. VARADHAN, Large Deviations and Applications KiYosi ITó, Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces Solitons in Mathematics and Physics Theory and Applications of Sequential Nonparametrics LÁszLó LovA'sz, An Algorithmic Theory of Numbers, Graphs and Convexity E. W. CHENEY, Multivariate Approximation Theory: Selected Topics JOEL SPENCER, Ten Lectures on the Probabilistic Method PAUL C. FIFE, Dynamics of Internal Layers and Diffusive Interfaces CHARLES K. CHUI, Multivariate Splines HERBERT S. WILF, Combinatorial Algorithms: An Update HENRY C. TUCKWELL, Stochastic Processes in the Neurosciences FRANK H. CLARKE, Methods of Dynamic and Nonsmooth Optimization ROBERT B. GARDNER, The Method of Equivalence and Its Applications GRACE WAHBA, Spline Models for Observational Data RICHARD S. VARGA, Scientific Computation on Mathematical Problems and Conjectures INGRID DAUBECHIES, Ten Lectures on Wave/ets STEPHEN F. MCCORMICK, Multilevel Projection Methods for Partial Differential Equations HARALD NIEDERREITER, Random Number Generation and Quasi-Monte Carlo Methods JOEL SPENCER, Ten Lectures on the Probabilistic Method, Second Edition CHARLES A. MICCHELLI, Mathematical Aspects of Geometric Modeling ROGER TEMAM, Navier—Stokes Equations and Nonlinear Functional Analysis, Second Edition GLENN SHAFER, Probabilistic Expert Systems PETER J. HUBER, Robust Statistica) Procedures, Second Edition J. MICHAEL STEELE, Probability Theory and Combinatorial Optimization WERNER C. RHEINBOLDT, Methods for Solving Systems of Nonlinear Equations, Second Edition J. M. CUSHING, An Introduction to Structured Population Dynamics TAI-PING LIU, Hyperbolic and Viscous Conservation Laws MICHAEL RENARDY, Mathematica) Analysis of Viscoelastic Flows GÉRARD CORNUÉJOLS, Combinatorial Optimization: Packing and Covering IRENA LASIECKA, Mathematica) Control Theory of Coupled PDEs J. K. SHAW, Mathematica) Principles of Optical Fiber Communications ZHANGXIN CHEN, Reservoir Simulation: Mathematica) Techniques in Oil Recovery ATHANASSIOS S. FOKAS, A Unified Approach to Boundary Value Problems MARGARET CHENEY AND BRETT BORDEN, Fundamentals of Radar Imaging ALAN C. NEWELL, PRANAB KUMAR SEN, Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Copyright © 1973 by the Society for Industrial and Applied Mathematics. 10987 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. ISBN 978-0-898710-07-6 n SULaJTL is a registered trademark. Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Contents Preface ......................................................... ix 1. Introduction 1.1. Preliminaries ............................................. 1.2. The sample d.f. as a stochastic process ........................ 1.3. The Brownian motion processen ............................. 1 2 4 2. Kolmogorov—Smirnov tests, finite-sample case 2.1. Distributions of one-sided Kolmogorov—Smirnov statistics....... 2.2. Distribution of the value of t at which max (FF (t) — t) occurs...... 2.3. Pyke's statistica ........................................... 2.4. Distributions of two-sided Kolmogorov—Smirnov statistics ...... 2.5. The probab lity that F(t) crosses a general boundary ............ 6 9 9 10 i13 3. Asymptotic theory of Kolmogarov—Smirnov tests 3.1. Limiting finite-dimensional distributions of the sample process ... 3.2. Weak convergente of the sample dl .......................... 3.3. Distribution of sup, (y(t)) .................................... 3.4. Distribution of sup, ly(t)j .................................... 3.5. The probability that y(t) crosses a general boundary ............ 3.6. Series expansions for the distribution functions of Kolmogorov— Smirnov statistics. ................ . 3.7. Powers of Kolmogorov—Smirnov tests ........................ 17 17 18 20 23 25 25 4. Cramér—von Mises tests 4.1. The test statistics .......................................... . 26 4.2. Exact results ...................................... 0 ....... 27 4.3. Components of Cramér—von Mises statistics ................... 28 4.4. Asymptotic distributions ................................... 30 4.5. Other statistics based on the sample d.f. .....32 32 5. Tests on the circle 5.1. Kuiper's statistic V„ ........................................ 5.2. Exact distribution of V ..................................... 5.3. Asymptotic distribution of V„ ................................ 5.4. Watson's statistic UM ....................................... 5.5. Exact results for U^ ........................................ 5.6. Asymptotic distribution of Un ................... 0 ........... 5.7. Other statistics for tests on the circle ......................... vii 33 34 35 36 37 37 39 Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php viii CONTENTS 6. Two-sample tests 6.1. Kolmogorov—Smirnov tests ...................... ........... 6.2. Limiting distributions of Kolmogorov—Smirnov statistics ........ 6.3. Cramér—von Mises statistica ................................. 6.4. Limiting distribution of W„ „ ....... ........................ 6.5. Two-sample tests on the circle ............................... 1 0 39 44 44 46 46 7. Tests based on the sample d.f. when parameters are estimated 7.1. Introductory .............................................. 47 48 7.2. Kolmogorov—Smirnov tests, finite-sample case .48 7.3. Weak convergence of the sample d.f. when parameters are estimated 53 .................. 54 7.4. Limiting distribution of Dn .............. 7.5. Cramér—von Mises tests .................................... 56 0 References...................................................... 59 Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php Preface This monograph is based on ten lectures given by me at a conference on tests based on the sample distribution function held at the State University of New York at Buffalo, in August/September 1971. These were developed from a course I gave white a Visiting Fellow at the Australian National University during the session 1970/71. The literature on the subject is now enormous and the choice of material for inclusion reflects my predilections as a methodological statistician concerned with the development of techniques which practical workers will find useful. My main objective has been to present a coherent body of theory for the derivation of the sampling distributions of a wide range of test statistics. Efforts have been made to unify the treatment; for example, I have tried to relate the derivations for tests on the circle and the two-sample problem to the basic theory for the onesample problem on the line. I have also placed much emphasis on the Markovian nature of the sample distribution function since this accounts for the remarkable elegance of many of the results achieved as well as the close relation with parts of the theory of stochastic processes. I have taken the opportunity to include in the sections on tests after parameters have been estimated some results obtained since the conference took place where these constitute an improvement over those presented at the time. Among omissions, the one I regret most is a discussion of the application outside the goodness-of-fit context of the techniques under study, particularly since my own interest in the area originated from the application of the Kolmogorov—Smirnov test to tests of serial correlation in time series analysis. The written version differs from the lectures as delivered in three main respects. First, I have had to leave out some informal comments on the relation of the theory presented to practical and theoretical work in a variety of fields which I hope helped to enliven the verbal presentation. Secondly, because of the need for precision of statement in a written record, the domination of ideas by algebraical detail in the monograph is inevitably greater than in the lectures. Thirdly, I have given no indication whatever of the benefit derived by all present, and particularly by me, from the vigorous discussion by the audience of the various problems raised. Also to be recalled are the excellent talks given by other participants on their own work in the general area covered by the conference. I mention these points here in order to ensure that the dry record I have written should not obscure completely the memory of a lively social occasion. 1 am extremely grateful to Professor Emanuel Parzen, Chairman of the Statistics Department at SUNYAB, for inviting me to give these lectures and for ix Downloaded 04/04/15 to 134.157.146.58. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php PREFACE his excellent organization of the conference facilities. Thanks are due also to the Conference Board of Mathematical Sciences for administrative support and to the National Science Foundation for finance. I am indebted to Martin Knott, Colin Taylor and Roger Heard for help in removing errors Erom the first draft. J. DURBIN