Bernard W. Lindgren - Statistical Theory: 22 (Chapman & Hall/CRC Texts in Statistical Science)

STATISTICAL THEORY
FOURTH EDITION

Bernard W. Lindgren

Library of Congress Cataloging-in-Publication Data

Lindgren, B.W. (Bernard William), 1924

Statisical theory/Bernard Lindgren. 4th ed.

p. cm.

Includes bibliographical references and index.

ISBN 0-412-04181-2

1. Mathematical statistics. I. Title.

QA276.L546

519.5dc20 93-1042

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International Standard Book Number 0-412-04181-2

Library of Congress Card Number 93-1042

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Like the earlier editions, this textbook is intended for a years course in the theory of statistics. No previous knowledge of statistics is assumed. However, a study of the theory is perhaps better motivated and more appreciated by students who have had some exposure to statistical methodology. I assume that the student has a good command of first-year calculus as well as some knowledge of multivariable calculus and at least an introduction to linear algebra.

In the seventeen years since the third edition, the body of knowledge we think of as statistical theory has of course expanded considerably, but the material covered in earlier editions remains important as a foundation, both for a practitioners better understanding of basic statistical methods and for a theoreticians further excursions into the theory. Thus, I have not changed the coverage in any significant way.

What I have done is to eliminate certain material that made the text seem more formidable than it wasmaterial that was usually passed over in our own teaching from the text. There is still ample material, however, for a course of three lectures per week. The major change in organization is the deferring of statistical decision theory (per se) to a final chapter, although the Bayesian approach is now introduced earlierin the chapters on estimation and testing, and before that, as one motivation for studying likelihood.

Without a background of measure theory, a mathematically rigorous treatment (definition-theorem-proof) is not feasible. For this reason I avoided stating formal theorems in the earlier editions. This had the effect of tending to hide important and useful information, so in this revision I have stated results as theorems, with the understanding that perhaps both statement and proof might be wanting in mathematical rigor.

In the process of rewriting I have naturally tried to clarify the exposition, and generally to make the text more accessible. And I have increased the number of problems for the student by about forty percent. Answers, for problems that have answers, are given at the back of the book. An instructors manual, with problem solutions and additional problems, will be available.

I am indebted to students and colleagues in the School of Statistics at the University of Minnesota whose reactions, comments, corrections, and criticismsbased on their experience with earlier editions and their classroom use of a preliminary version of this fourth editionhave been most helpful. I am especially indebted to Ms. Wei Shen, whose careful reading of the text and solving of all of the problems have eliminated numerous errors.

Bernard W. Lindgren

An experiment is doing something or observing something happen under certain conditions, resulting in a final state of affairs or outcome. The experiment may be physical, chemical, social, medical, industrial, educational, etc. It may be conducted in the laboratory or in real-life surroundingsin the factory, the economy, the hospital, the classroom.

An experiment may be one whose outcome is completely determined by what one knows of the conditions under which it is carried out. In practice, however, one often finds that the outcome varies from one trial to another. This will happen when there are conditions or factors affecting the outcome that cannot be measured or taken into account or (perhaps) even identified. We say that such experiments involve chance, and refer to them as experiments of chance. In such cases the usual type of deterministic model, involving equations of motion and/or equations of state that embody physical or other laws, is inadequate for describing or predicting what will happen at a given trial. A different type of mathematical structure is called fora probability model.

For understanding how things work, for aid in making decisions or plans or predictions, we need to know the appropriate model. Statistical practice involves conducting one or more trials of an experiment and analyzing the results for evidence about the model. Drawing conclusions about the model from what we see in experimental data is statistical inference. Before taking up the topic of inference, however, we need to study probability models.

The basic ingredients of a probability model are a sample space, a collection of events, and probabilities of events. The sample space of an experiment is simply the set of possible outcomes, and events are subsets of the sample space. We begin with a review of some useful devices for counting outcomes, and of some elementary algebra of sets.

Counting the outcomes of an experiment is easy when we can make a list of themone that is not too long. But in many instances this is simply not feasible. If the experiment can be thought of as a composition of simpler experiments, we can count outcomes by appealing to the following simple principle:

COUNTING PRINCIPLE