Elementary Decision Theory

Herman Chernoff, Lincoln E. Moses

• Publisher: Dover Publications
• Year: 2012
• ISBN:
  9780486143774
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

`The text is very clearly written [with] many illustrative examples and exercises [and] should be considered by those instructors who would like to introduce a more modern (and a more logical) approach in a basic course in statistics.` —Journal of the American Statistical Association
This volume is a well-known, well-respected introduction to a lively area of statistics. Professors Chernoff and Moses bring years of professional expertise as classroom teachers to this straightforward approach to statistical problems. And happily, for beginning students, they have by-passed involved computational reasonings which would only confuse the mathematical novice.
Developed from nine years of teaching statistics at Stanford, the book furnishes a simple and clear-cut method of exhibiting the fundamental aspects of a statistical problem. Beginners will find this book a motivating introduction to important mathematical notions such as set, function and convexity. Examples and exercises throughout introduce new topics and ideas.
The first seven chapters are recommended for beginning courses in the basic ideas of statistics and require only a knowledge of high school math. These sections include material on data processing, probability and random variables, utility and descriptive statistics, uncertainty due to ignorance of the state of nature, computing Bayes strategies and an introduction to classical statistics. The last three chapters review mathematical models and summarize terminology and methods of testing hypotheses. Tables and appendixes provide information on notation, shortcut computational formulas, axioms of probability, properties of expectations, likelihood ratio test, game theory, and utility functions.
Authoritative, yet elementary in its approach to statistics and statistical theory, this work is also concise, well-indexed and abundantly equipped with exercise material. Ideal for a beginning course, this modestly priced edition will be especially valuable to those interested in the principles of statistics and scientific method.

CHAPTER 1

Introduction

1. INTRODUCTION

Beginning students are generally interested in what constitutes the subject matter of the theory of statistics. Years ago a statistician might have claimed that statistics deals with the processing of data. As a result of relatively recent formulations of statistical theory, today’s statistician will be more likely to say that statistics is concerned with decision making in the face of uncertainty. Its applicability ranges from almost all inductive sciences to many situations that people face in everyday life when it is not perfectly obvious what they should do.

What constitutes uncertainty? There are two kinds of uncertainty. One is that due to randomness. When someone tosses an ordinary coin, the outcome is random and not at all certain. It is as likely to be heads as tails. This type of uncertainty is in principle relatively simple to treat. For example, if someone were offered two dollars if the coin falls heads, on the condition that he pay one dollar otherwise, he would be inclined to accept the offer since he “knows” that heads is as likely to fall as tails. His knowledge concerns the laws of randomness involved in this particular problem.

The other type of uncertainty arises when it is not known which laws of randomness apply. For example, suppose that the above offer were made in connection with a coin that was obviously bent. Then one could assume that heads and tails were not equally likely but that one face was probably favored. In statistical terminology we shall equate the laws of randomness which apply with the state of nature.

What can be done in the case where the state of nature is unknown? The statistician can perform relevant experiments and take observations. In the above problem, a statistician would (if he were permitted) toss the coin many times to estimate what is the state of nature. The decision on whether or not to accept the offer would be based on his estimate of the state of nature.

One may ask what constitutes enough observations. That is, how many times should one toss the coin before deciding? A precise answer would be difficult to give at this point. For the time being it suffices to say that the answer would depend on (1) the cost of tossing the coin, and (2) the cost of making the wrong decision. For example, if one were charged a nickel per toss, one would be inclined to take very few observations compared with the case when one were charged one cent per toss. On the other hand, if the wager were changed to $2000 against $1000, then it would pay to take many observations so that one could be quite sure that the estimate of the state of nature were good enough to make it almost certain that the right action is taken.

It is important to realize that no matter how many times the coin is tossed, one may never know for sure what the state of nature is. For example, it is possible, although very unlikely, that an ordinary coin will give 100 heads in a row. It is also possible that a coin which in the long run favors heads will give more tails than heads in 100 tosses. To evaluate the chances of being led astray by such phenomena, the statistician must apply the theory of probability.

Originally we stated that statistics is the theory of decision making in the face of uncertainty. One may argue that, in the above example, the statistician merely estimated the state of nature and made his decision accordingly, and hence, decision making is an overly pretentious name for merely estimating the state of nature. But even in this example, the statistician does more than estimate the state of nature and act accordingly. In the $2000 to $1000 bet he should decide, among other things, whether his estimate is good enough to warrant accepting or rejecting the wager or whether he should take more observations to get a better estimate. An estimate which would be satisfactory for the $2 to $1 bet may be unsatisfactory for deciding the $2000 to $1000 bet.

2. AN EXAMPLE

To illustrate statistical theory and the main factors that enter into decision making, we shall treat a simplified problem in some detail. It is characteristic of many statistical applications that, although real problems are too complex, they can be simplified without changing their essential characteristics. However, the applied statistician must try to keep in mind all assumptions which are not strictly realistic but are introduced for the sake of simplicity. He must do so to avoid assumptions that lead to unrealistic answers.

Example 1.1. The Contractor Example. Suppose that an electrical contractor for a house knows from previous experience in many communities that houses are occupied by only three types of families: those whose peak loads of current used are 15 amperes (amp) at one time in a circuit, those whose peak loads are 20 amp, and those whose peak loads are 30 amp. He can install 15-amp wire, or 20-amp wire, or 30-amp wire. He could save on the cost of his materials in wiring a house if he knew the actual needs of the occupants of that house. However, this is not known to him.

One very easy solution to the problem would be to install 30-amp wire in all ho...

Table of contents

• Title Page
• Copyright Page
• Dedication
• Preface
• Acknowledgments
• Table of Contents
• CHAPTER 1 - Introduction
• CHAPTER 2 - Data Processing
• CHAPTER 3 - Introduction to Probability and Random Variables
• CHAPTER 4 - Utility and Descriptive Statistics
• CHAPTER 5 - Uncertainty due to Ignorance of the State of Nature
• CHAPTER 6 - The Computation of Bayes Strategies
• CHAPTER 7 - Introduction to Classical Statistics
• CHAPTER 8 - Models
• CHAPTER 9 - Testing Hypotheses
• CHAPTER 10 - Estimation and Confidence Intervals
• APPENDIX A - Notation
• APPENDIX B1
• APPENDIX C1
• APPENDIX D1
• APPENDIX E1
• APPENDIX F1 - Remarks About Game Theory
• Partial List of Answers to Exercises
• Index