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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.
