DISCERNIBLE PERIODS IN THE HISTORICAL DEVELOPMENT OF STATISTICAL INFERENCE by RICHARD WAYNE GOBER A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the School of Connnerce and Business Administration in the Graduate School of the University of Alabama UNIVERSITY, ALABAMA 1967 ACKNOWLEDGMENTS Special acknowledgment is given to my dissertation committee, Dr. John P. Gill, Chairman, and Dr. A. Lee Cobb, for their skilled guidance, valuable suggestions, and encouragement; Dro Albert Drake and Dr. Tom D. Moore, for their assistance in preparing the final copye To my wife and family I am grateful for their patience and approbation. ii CONTENTS Page ACKNOWLEDGMENTS ii Chapter I. INTRODUCTION 1 Purpose and Scope of the Study •••••.. 1 Background of the Study •• 0 . 0 2 Limitations and Procedures ••• e 4 II. THE PERIOD OF INDUCTIVE STATISTICAL INFERENCE. 6 The Principle of Inverse Probability 6 Origination by Bayes ••• e ••• 6 Enunciation and Generalization by Laplace 13 Controversy Relating to the Principle •. 16 Criticism of the Principle . 16 Defense of the Principle . 18 Repudiation by Fisher 20 Sampling Distributions • • • ff . .. 23 The Normal Distribution 24 Discovery of the Normal Curve ••.•. 24 The Normal Sampling Distribution 24 Use of the Modulus and the Probable Error. 26 The Chi-Square Distribution. 28 Discovery by Helmert 28 The Chi-square Test of Goodness of Fit •••• 29 Contingency Tables •.•••••• 31 Controversy Relating to Degrees of Freedom 33 Student's Distribution 34 Discovery by Gosset •••• 34 Extended Applicatiorr by Fisher 36 iii Chapter Page Fisher's z Distribution o ~ $ 37 Fisher's z Transformation o 38 The Distribution of z • 39 Analysis of Variance 41 III. THE PERIOD OF CLASSICAL STATISTICAL INFERENCE. 44 Statistical Interval Estimation. 45 The Problem of Interval Estimation 45 Fiducial Probability • o c • 46 Confidence Intervals .•. o 48 Differences in Formulation of Theories 51 Statistical Hypotheses Testing 54 Statement of the Neyman-Pearson Theory 55 Development of the Neyman-Pearson Theory 58 The Likelihood Ratio Criterion 59 Additional Principles for Test Selection 62 The Power of Classical Tests ..••• 64 Fisher's Reaction to the Neyman-Pearson Theory 66 The Behrens...:. Fisher Problem" • . • . 69 Controversy Over the Behrens-Fisher Solution 71 The Controversy Evolves Into a Clash •.•.• 73 IV. THE PERIODS OF NONPARAMETRIC AND DECISION-THEORETIC STATISTICAL INFERENCE. • . • • . ••• 76 Nonparametric Statistical Inference 76 Nonparametric Test Problems 80 The Sign Test •• o 81 Order Statistics .•... 85 Rank Tests 90 Randomization and Independence 92 Power of Nonparametric Tests 95 Decision-Theoretic Statistical Inference 98 Basic Concepts of the Theory 99 Historical Development ••• 102 iv Chapter Page V. THE PERIOD OF BAYESIAN STATISTICAL INFERENCE . 107 Beginning and Basis .. 108 Bayes' Theorem ..... 109 A Priori Distributions ... 109 Frequentist Interpretation •• 0 110 Bayesian Interpretation •. 112 Epistemological Approach • 112 Personal or Individualistic Approach .. 114 Likelihood Functions 118 Bayesian Analysis ... 120 Confidence Distributions 122 Robustness • 123 Tolerance Regions .. • • 0 124 VI. SUMMARY AND CONCLUSIONS . • . 126 An Overview of the Development . • . • . • 126 Discernible Patterns in the Development • o o • r 132 A Priori Information ...•..........• 132 Interaction Between Theory and Application ...... 134 The Pattern of Controversy . • • • . 136 Increasing Mathematical Content . • 137 Conclusions . • 0 138 APPENDIX 140 BIBLIOGRAPHY . 148 V CHAPTER I INTRODUCTION Purpose and Scope of the Study The purpose of this study is to trace the historical develop­ ment of that part of modern statistical procedures known as statistical inference. Although the application of statistical methods is concerned more than ever with the study of great masses of data, percentages, and columns of figures, statistics has moved far beyond the descriptive stage. Using concepts from mathematics, logic, economics, and psychol­ ogy, modern statistics has developed into a designed "way of thinking" about conclusions or decisions to help a person choose a reasonable course of action under uncertainty. The general theory and methodology is called statistical inference. Statistical inference, roughly described, is that part of sta­ tistics in which quantitative measures of uncertainty are employed. Generally, statistical inference is a way to formalize the many mental assumptions, facts, and goals that go into making a decision under uncertainty, where a decision is made by following certain logical principles to reduce the chance of error and inconsistent action. Modern textbook writers avoid the historical approach in favor of a logical presentation of statistical practices and techniques in current use. As a result, the student of statistics is denied the opportunity to learn about the evolution of a flourishing and expanding 1 2 branch of statistical knowledge. To provide this historical perspective, the scope of this study will include salient contributions of the past two hundred years. Background of the Study The problem of using the facts of experience to infer general conclusions is common to all branches of science seeking to establish laws in terms of which a rational explanation may be given observable phenomena. The facts of experience that provide the impetus for this problem of inductive reasoning do not unrelentingly lead to categorical conclusions. The uncertainty characteristic of drawing conclusions or making decisions on the basis of limited information is analogous to the uncertainty characteristics of the outcomes of games of chance. The success which seventeenth century mathematicians had in dealing with the uncertainty characteristics of the outcomes of games of chance made probability a logical basis for handling the uncertainty character­ istics of the conclusions or decisions drawn from limited information. Such early attempts to use probability inductively were the beginning of the use of inductive inference in statistics. Various periods are discernible in the development of statis­ tical inference, each of which has a very definite beginning but no end. Until the latter part of the nineteenth century the concept was that of a loosely defined set of ideas expressed in terms of justifying propositions on the basis of data. The problem was that of inductive inference which introduced methods of making inferences about the general from the study of the particular. After the turn of the cen­ tury a measurement about the uncertainty of an inference was added. In the 1920's and 1930's the number of persons competent to develop 3 new statistical theory increased rapidly, and the concept of inductive behavior appeared. The same applies for concepts of tests of statis­ tical hypotheses, of errors of the first and second kind, of power of the tests, and of interval estimates" The emphasis was now on the importance of decisions or actions, as opposed to assertions or con­ clusions, in the face of uncertaintyo In 1936, attention was focused on the restrictive assumptions about the functional form of the dis- tribution of the population sampled. The fact that statistical methods were not limited to inferences concerning certain parameters resulted in the development of the nonparametric or distribution-free approach to statistical inference. During and shortly after World War II, sta­ tistical inference advanced with a decision-theoretic, or economic, outlook. The decision-theoretic formulation made use of sequential analysis to add the concept of risk to the decisions or actions approacho The latest period in the development of statistical inference started in the 1950's as the influence of the decision-theoretic framework com­ bined with the notion of personal probability and a procedure for probability calculation, Bayes' theorem, to form the concept of Bayesian statistical inference. The steadily increasing importance of statistical inference in such areas as government, industry, and education, and the rapid growth in the theory and application of statistical methods designed for this purpose underscore the desirability of an account of the historical development of statistical inferenceo By increasing the efficiency with which the scientific method can be applied, research in the use of statistical inference has made 'a lasting contribution to the solution of problems in many divergent fields~ Some of these problems, such 4 as predicting the performance of a complex new missile system, ascer­ taining the effectiveness of new vaccines, the evaluation of the possible effects of smoking on health, and the dete-r'mination of public opinion on various national and local issues,) have gained much publicity, Other problems, to which statistical inference has made a notable contribution, have received less publicity~ Some of the problems, however, no less significant with respect to the implications of their solution, include the following: discovering how galaxies of stars change shape as they grow old, forecasting the change in the long distance traffic that would follow an increase in toll rates, control of the quality of manufactured articles, and construction of mathematical models for the purpose of explaining human behavior. Limitations and Procedures When sketching a history of scientific thought, such as sta­ tistical inference, and trying to indicate periods of birth and develop­ ment of particular ideas, there is invariably