Mathematical Statistics

MATHEMATICAL STATISTICS By S. S. WILKS PRINCETON UNIVERSITY PRESS Princeton, New Jersey 1947 Copyright, 1943, by PRINCETON UNIVERSITY PRESS PREFACE Moat of the mathematical theory of statistics In Its present state has been developed during the past twenty years. Because of the variety of scientific fields In which statistical problems have arisen, the original contributions to this branch of applied mathematics are widely scattered In scientific literature. Most of the theory still exists only In original form. During the past few years the author has conducted a two-semester course at Princeton University for advanced undergraduates and beginning graduate students in which an attempt has been made to give the students an Introduction to the more recent develop ments in the mathematical theory of statistics. The subject matter for this course has been gleaned, for the most part, from periodical literature. Since it is impossible to cover in detail any large portion of this literature in two semesters, the course has been held primarily to the basic mathematics of the material, with just enough problems and examples for illustrative and examination purposes. Except for Chapter XI, the contents or the present set of notes constitute the basic subject matter which this course was designed to cover. Some of the material in the authors Statistical Inference 1937 has been revised and Included. In writing up the notes an attempt has been made to be as brief and concise as possible and to keep to the mathematics with a minimum of excursions into applied mathematical statistics problems. An important topic which has been omitted is that of characteristic functions of random variables, which, when used in Fourier Inversions, provide a direct and powerful method of determining certain sampling distributions and other random variable distribu tions. However, moment generating functions are used they are more easily understood by students at this level and are almost as useful as characteristic functions as far as actual applications to mathematical statistics are concerned. Many specialized topics are omitted, such as intraclass, tetrachoric and other specialized correlation problems, aeml-lnvariants, renewal theory, the Behrens - Fisher problem, special transformations of population parameters and random variables, sampling from Poisson populations, etc. It is the experience of the author that an effective way for handling many of these specialized topics is to formulate them as problems for the students. If and when the present notes are revised and issued in permanent form, such problems will be Inserted at the ends of sections and chapters. In the meantime, criticisms, suggestions, and notices of errors will be gratefully received from readers. Finally, the author wishes to express his indebtedness to Dr. Henry Scheffe, Mr. T. W. Anderson, Jr. and Mr. D. F. Votaw, Jr. for their generous assistance in pre paring these notes. Most of the sections in the first seven chapters and several sections in ChapteSX and XI were prepared by these men, particularly the first two. Thanks are due Mrs. W. M. Weber for her painstaking preparation of the manuscript for lithoprinting. S. S. Wilks. Princeton, New Jersey April, TABLE OF CONTENTS CHAPTER I, INTRODUCTION 1 CHAPTER II. DISTRIBUTION FUNCTIONS 52.1 Cumulative Distribution Functions 5 2.11 IMivariate Case 5 2.12 Blva rlate Case 852.13 k-Variate Case 11 2.2 Marginal Distributions 12 52.3 Statistical Independence 13 52.1 Conditional Probability 15 52.5 The Stleltjes Integral 17 52.51 Univarlate Case 17 52.52 Blvarlate Case 20 52.53 k-Variate Case 21 52.6 Transformation of Variables 23 2.61 Unlvariate Case 2k 2.62 Blvarlate Case 2k 52.63 k-Vtetrlate Case 28 52.7 Mean Value 29 52.71 Univarlate Case Tchebychef f f s Inequality 30 52.72 Bivariate Case 31 52.73 k-Variate Case 32 52.71 Mean and Variance of a Linear Combination of Bandom Variables 33 52...