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william a. small (jb ► P. 0. BOX 122-A Sl Tennessee Polytechnic Institute COOKEVILLE, TENNESSEE ^ j \ ci Q [ 6 * f u> »<- \ % STATISTICAL DECISION THEORY McGraw-Hill Series in Probability and Statistics David Blackwell, Consulting Editor Bharucha-Reid. Elements of the Theory of Markov- Processes and Their Applications Graybill. Introduction to Linear Statistical Models, Volume I Wadsworth and Bryan. Introduction to Probability and Random Variables Weiss. Statistical Decision Theory . i STATISTICAL DECISION THEORY LIONEL WEISS ASSOCIATE PROFESSOR DEPARTMENT OF INDUSTRIAL AND ENGINEERING ADMINISTRATION CORNELL UNIVERSITY New York Toronto London 1961 McGRAW-HILL BOOK COMPANY, INC. STATISTICAL DECISION THEORY Copyright © 1961 by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers. Library of Congress Catalog Card Number 61-11132 69068 PREFACE At the present time there are very few textbooks on statistical decision theory, and there is apparently no textbook that gives a reasonably complete discussion of decision theory at an intermediate mathematical level. The author hopes that this textbook will help to fill the gap. It has as formal mathematical prerequisites only calculus through partial differentiation and multiple integration, plus some elementary facts about the use of determinants in solving linear equations. A few sections of the text are demanding on a student with the amount of mathematical maturity usually implied by a year course in calculus. The derivation of the Wald sequential decision rule in Chapter 7 is the most important example. The student will be familiar with all the mathematical tools used, but because of the length of the development, the instructor will have to keep the student from being overwhelmed by the details. Although the purpose of the text is to teach statistical decision theory, a substantial proportion of the text is devoted to a simple and relatively brief discussion of probability theory. This makes the text self- contained for those students who have not previously studied probability theory. Students who have had a course in probability theory should find the discussion of this theory a useful review, since the emphasis is on those topics of probability theory most useful in statistical decision theory. Some of the topics discussed in the text are unusual for courses in the general field of statistics. Two examples are linear programming and making a sequence of nonsampling decisions over time. The author feels that these topics belong in a course on statistical methods. For reasons explained in Chapter 5, the loss is made to depend on the decision chosen and on chance variables that will be observed after the decision has been chosen, rather than on the decision chosen and the distribution of the chance variables. PREFACE VI Chapter 9, the last chapter, introduces conventional statistical methods as special cases of statistical decision theory. The examples discussed in Chapter 9 seem a step further from practical problems than the examp es discussed in earlier chapters. This seems to be an inherent property of the problems discussed in conventional statistical theory. .. , In a text at this level, it does not seem desirable to include detailed statements about which researchers are responsible for the various results described. Statistical decision theory is the creation of the late A Wald - the proof that the Wald sequential rule is a Bayes decision rule is due to A. Wald and J. Wolfowitz; and the simplex method is due to G. B. Dantzig. . , How long a course should be devoted to the text depends on the maturity of the students and on the taste and style of the instructor The author has used forty-five hours of lectures for a brief review of Chapters 1 to 4 and a thorough discussion of the remaining chapters, including the discussion of many numerical examples. This will seem to be a s low pace to many instructors, who will be able to cover the whole text fairly thoroughly in a forty-five-hour course. . Chapter 9 serves as an introduction to conventional statistical theory, and it is possible to use the text for one semester and then to use a text on conventional statistical methods for the. second semester of a year course. The author is indebted to Professor Sir Ronald A. Fisher, F.R.S., Cambridge, and to Messrs. Oliver & Boyd Ltd., Edinburgh, for per¬ mission to reprint the tables of the chi-square and the t distribution from “Statistical Methods for Research Workers.” The author is also indebted to Professor G. W. Snedecor and the Iowa State University Press, Ames, Iowa, for permission to reprint the table of the ^distribution from “Statistical Methods.” For valuable typing assistance, the author is grateful to Miss Ruth Ritchie of the University of Virginia, Mrs. David Freeman, formerly of Ithaca, N.Y., Mrs. Robert Berlin of Syracuse, N.Y., and to Rhoda Weiss. Lionel Weiss.