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STATISTICAL MODELS AND ANALYSIS IN AUDITING A STUDY OF STATISTICAL MODELS AND METHODS FOR ANALYZING NONSTANDARD MIXTURES OF DISTRIBUTIONS IN AUDITING Pane! on Nonstandard Mixtures of Distributions Committee on Applied and Theoretical Statistics Board on Mathematical Sciences Commission on Physical Sciences, Mathematics, and Resources National Research Council NATIONAL ACADEMY PRESS Washington, D. C. 1988

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NOTICE: Lee project that is the subject of this report was approved by the Govenung Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of He cornnuttee responsible for the report were chosen for their special competences and with regard for appropriate balance. Lois report has been reviewed by a group other than the authors according to procedures approved by a Report Review Committee consisting of members of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The National Academy of Sciences is a pnvate, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Frank Press is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the choicer of the National Academy of Sciences as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Robert M. White is president of the National Academy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts ~ . . ~ . . . ~ ~ . ~ under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Samuel O. Tier is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad commuruty of science and technology with the Academy's purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies deterrn~ned by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Instinlte of Medicine. Dr. Frank Press and Dr. Robert M. White are chairman and vice chairrnas~, respectively, of the National Research Council. Support for this project was provided by the Treasury Department and the Defense Logistics Agency, and by core funds for the Board on Mathematical Sciences, National Research Council, provided by the National Science Foundation, the Air Force Office of Scientific Research, the Army Research Office, the Department of Energy and Be Office of Naval Research. Available from Board on Mathematical Sciences, National Research Council 2101 Constitution Ave., NW Washington, D.C. 20418 Printed in the L'nited States of America

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PANEL ON NONSTANDARD MIXTURES OF DISTRIBUTIONS DONALD GIJIXRE, University of (~lifomia, Los Angeles, C~innan Z. W. BIRNBAUM, University of Washington, Seattle W~FRID J. DIXON, University of California, Los Angeles STEPHEN E. FIENBERG, Carnegie-Mellon University, Pittsburgh JANE F. GENTLEMAN, University of Waterloo, Waterloo, Canada JAMES M. LANDWEHR, AT&T Bell Laboratories, Murray Hill NANCY R. MANN, University of California, Los Angeles PAUL W. MIELKE, Colorado State University, Fort Collins JOEW NE.T=? University of Georgia, Athens DONALD M. ROBERTS, University of Illinois, Urbana JOHN VAN RYZIN*, Columbia University, New York H. TAMURA+, University of Washington, Seattle ARTHUR J. WILBURN, AJ. Wilburn Associates, Rock~rille JAMES S. WILLIAMS, Colorado State University, Fort Collins * Deceased, March 6, 1987 + Appointed, August 30, 1985 ~e

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COMMITTEE ON APPLIED AND THEORETICAL STATISTICS RONALD PYKE, University of Washington, Chairman RALPH A. BRADLEY, University of Georgia MORRIS De GROOT, Camegie-Mellon Universin,r WILLIAM EDDY, Carnegie-Mellon University JOHN A. HARTIGAN, Yale University LUCEN LE CAM, University of California at Berkeley GARY C. McDONALD, General Motors Research Labs JOHN RICE, University of California at San Diego JOAN R. ROSENBLA=, National Bureau of Standards JEROME SACKS, University of Illinois J. SETHURAMAN, Florida State University MICHAEL J. STEELE, Princeton University GEORGE C. TWO, University of Chicago IV

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BOARD ON MATHEMATICAL SCIENCES PHILLI:P A. GR1~l'1 HISS Duke University, Chapman PETER J. BICKEL, University of California, Berkeley HERMAN CHERNOFF, Harvard University RONALD G. DOUGLAS, SUNY Stony Brook E. F. lNFANlE, University of Minnesota WILLIAM JACO, Oklahoma State University JOSEPH J. KOHL, Princeton University CATHLEEN S. MORAWEIZ, New Yolk University ALAN NEWELL, University of Arizona GUIDO WEISS, Washington University SHMUEL WINOGRAD, IBM Corporation Ex-O~cio Member: RONALD PYKE (Chairman CATS), University of Washington NRC Staff LAWRENCE H. COX, Staff Director SEYMOUR SELIG, Staff Officer ROBERT SMYT~, Staff Officer WILLIAM G. ROSEN, Staff Officer ROSE M. KOPERA, Staff Assistant CATHY RICHARDSON, Secretary v

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COMMISSION ON PHYSICAI, SCIENCES, MATHEMATICS, AND RESOURCES NORMAN HACKERMAN, Robert A. Welch Foundation, Chapman GEORGE F. CARRIER, Harvard University DEAN E. EASTMAN, IBM, T. J. Watson Research Center FRYE ANNE FOX, University of Texas GERHART FREDLANDER, Brookhaven National Laboratory LAWRENCE W. FUNKHOUSER, Chevron Corporation (retired) PHILLIP A. GRWElTHS, Duke University J. ROSS MACDONALD, University of North Carolina, Chapel Hill CHARLES J. MANKIND, Oklahoma Geological Survey PERRY L. McCARTY, Stanford University JACK E. OLIVER, Comell University JEREMIAH P. OSTRIKER, Princeton University Observatory WILLIAM D. PHILLIPS, Mallinckrodt, Inc. DENIS J. PRAGER, Mac Arthur Foundation DAVID M. RAUP, University of Chicago RICHARD J. REED, University of Washington ROBERT E. SEVERS, University of Colorado LARRY L. SMARR, National Center for Supercompunng Applications EDWARD C. STONE, JR., California Institute of Technology KARL K. WREKIAN, Yale University GEORGE W. WET~RILL, Carnegie Institute of Washington IRVING WLADAWSKY-BERGER, IBM Data Systems Division RAPHAEL G. KASPER, Executive Director LAWRENCE E. McCRAY, Associate Executive Director Vl

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PREFACE The Committee on Applied and Theoretical Statistics (CATS) senses under the National Research Council's Board on Mathematical Sciences. CATS was created In 1977 in response to concerns about the uses of statistics in applications to the mathematical and physical sciences. The mandate of the Committee includes the foDow~ng: To maintain the awareness and promote We health of the discipline of statistics. To provide a base within the National Research Council for evaluating applications of statistics to the physical and computing sciences. To encourage the effective use of statistics in physical and computer science. To identify crucial questions in theoretics statistics whose solutions could have major consequences for applied statistics. To initiate studies, including interdisciplinary efforts, on problems of national and scientific importance. The first activity of the Committee was to hold a meeting at the Airlie House, Virginia, on February 9-11, 1978, involving both statisticians and scientists from the physical and earth sciences. Seventeen important problem areas for future emphasis were identified, and brief reports of their descriptions were prepared. These included for example the problems of large data bases, statistical computation, risk assessment and complex modeling. In the ten years that CATS has been in operation, a number of these and other topics have been considered for further study. There remain many other key areas where need for concerted attention remains. Examination of the state of the interface between statistics and auditing was one of the first interdisciplinary projects undertaken. On October 1, 1980, the Panel on Nonstandard Mixtures of Distributions was appointed under the chairmanship of Dr. Donald Guthne. The original purpose of the panel was "to review the existing general methodology of estimation procedures for mixtures of distributions, as well as methodology specific to applications in engineering and the physical sciences, the health sciences, and auditing and accounting." The members of the Panel constituted three subpanels representing the three general subject areas mentioned in the preceding quote. The subpanel for accounting and auditing included S. E. Fienberg, I. Neter, D. M. Roberts and A. J. Wilbum. Meetings and discussions were held by the subpanels and by He Panel as a whole dunug 1981. Preliminary outlines for a report were assigned and some draft material was discussed at a meeting of subpanel editors in 1983. Subsequently, an interim decision was made by CATS to restrict the focus of the report to the important special problems a. V11

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conceming nonstandard mixtures mat arise within Be area of statistical methodology in auditing. It was also decided to expand He subpanel for accounting and audimng. To this end, Dr. H. Tamura was added to the Pane! on August 30, 1985 so tot Us expense in auditing come ~ used to assist in the completion of the Report The survey of existing methodologies, Chapter Il. and He post-1981 updating of the subpanel's annotated bibliography. Chapter IV, have been prepared by Dr. Tamura, who has undertaken the overall responsibility for He report since his appointment to the Panel in 1985. The present report is He result of the efforts of many. In particular, I would express my gratitude to me chairman and members of He Panel as weld as to the members of CATS dung the term of this project. The completion of this project may not have been possible without the determined oversight of the previous chair of CATS, Ralph Bradley, who provided detailed comments to all of the preliminary drafts. I would also gratefully acknowledge He support of He staff of the Board of Mathematical Sciences under He direction of Frandc Gilfea~er. Ronald Pyke, Mailman of CATS, July, 1987 ~e v'''

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ACKNOWLEDGMENTS The preparation of dais report has benefited gready fiom the comments and suggestions of many people who reviewed several easier Crafts of He Report. Moreover, others have graciously provided their expertise to clarify technical issues as they arose. ~ addition to past and present members of CATS and Be members of Me Panel, we wish to express our appreciation to Wm. F. Felix, University of Arizona, I. Godfrey, University of Georgia, L. Headl, University of Washington, K. W. Stringer, New York University, and R. Bartyczak, T. Lieb, R. Weakland and D. Wilt of the Internal Revenue Service. Their considerable assistance is gratefully acknowledged.

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CONTENTS I. INTRODUCTION II. STATISTICAL MODELS AND ANALYSES IN AUDITING 1. The Beginnings Definitions and Notations Error Disuibui~ons of Audit Populations -Empirical Evidence 4. The Performance of Estimator Commonly Used for Human Populations When Applied to Accounting Populations 5. Confidence Bounds Using Attribute Sampling Theory 6. Over Developments for me Analysis of Dollar Unit Sample Data 7. Bayesian Models for He Analysis of Audit Data 8. Numerical Examples III. SUMMARY AND RECOMMENDATIONS IV. BIBLIOGRAPHY OF STATISTICAL PRACTICE IN AUDITING 1. Alphabetical Annotated Bibliography 2. Chronological Bibliography x Hi 8 8 12 14 21 32 37 38 47 55 60 60 84

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