Sequential Analysis

Abraham Wald

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486783239
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

In 1943, while in charge of Columbia University's Statistical Research Group, Abraham Wald devised Sequential Design, an innovative statistical inference system. Because the decision to terminate an experiment is not predetermined, sequential analysis can arrive at a decision much sooner and with substantially fewer observations than equally reliable test procedures based on a predetermined number of observations. The system's immense value was immediately recognized, and its use was restricted to wartime research and procedures. In 1945, it was released to the public and has since revolutionized many aspects of statistical practice.
This book is Professor Wald's own description of the system. Part I contains a discussion of the general theory of the sequential probability ratio test, with comparisons to traditional statistical inference systems. Part II discusses applications that illustrate the general theory and raise points of theoretical interest specific to these applications. Part III outlines a possible approach to the problem of sequential multi-valued decisions and estimation. All three sections can be understood by readers with only a background in college algebra and a first course in calculus. Mathematical derivations of somewhat intricate nature appear in the appendix.
Sequential Analysis offers statistical researchers a time- and money-saving approach, introduces students to one of the major systems in contemporary use, and presents those already acquainted with the system with valuable background information.

PART I. GENERAL THEORY

Chapter 1. ELEMENTS OF THE CURRENT THEORY OF TESTING STATISTICAL HYPOTHESES

1.1 Random Variables and Probability Distributions

1.1.1 Notion of a Random Variable

The outcome of an experiment or the reading of a measurement is usually a variable quantity or, more briefly, a variable, since generally it can take different values. For example, repeated measurements on the length of a bar will yield, in general, different values. Frequently, it will be possible to make probability statements concerning the outcome of an experiment or the reading of a measurement. Consider, for example, the experiment consisting of the throw of a die whose sides are numbered from 1 to 6. Here the outcome of the experiment may be any integral number from 1 to 6. Various probability statements regarding the outcome of the experiment can be made. For example, the probability that the outcome will be equal to 5 is equal to

, or the probability that the outcome will be less than 4 is equal to

, and so forth. Probability statements can also be made about the outcome of the following experiment: Suppose that an individual is selected at random from a group of 1000 individuals and that his height is then measured. The probability that the height of the selected individual will be less than 68 inches is equal to

times the number of individuals in the group whose heights are less than 68 inches.

A variable x is called a random variable if for any given value c a definite probability can be ascribed to the event that x will take a value less than c. A general class of experiments where the outcome is a random variable in the sense of the above definition may be described as follows. Consider a class of N objects (or individuals) and some measurable characteristic of these objects, such as weight, diameter, or hardness. Suppose that the value x of this characteristic varies from object to object in the class. The experiment consists in selecting at random one object from the class of N objects, and then measuring the value x of the characteristic of the selected object. Random selection is selection of an object in such a way that each object in the class of N objects has an equal chance of being chosen. The outcome x of such an experiment is a random variable, since a probability can be ascribed to the event that x will take a value less than c, for any given value c. This probability is, in fact, equal to N_c/N, where N_c is the number of objects in the class for which the characteristic undei consideration has a value less than c. An interesting special case is that in which the characteristic under consideration can take only two values. Such a situation arises, for instance, in the case of a manufactured product where each unit is classified in one of two categories: defective or non-defective. We shall ascribe the value 0 to a nondefective unit and the value 1 to a defective unit. Then the characteristic under consideration, i.e., the characteristic of being ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Introduction
• Part I. General Theory
• PART II. Application of the General Theory to Special Cases
• PART III. The Problem of Multi-Valued Decisions and Estimation
• Appendix
• Index