Mathematical Statistics

The present book is based on our experience, extending over a period of about fifteen years of teaching Mathematical Statistics to B.A (Hons) and M.A. classes of the University Delhi. During this period, a number of excellent books on Mathematical Statistics have appeared, but, surprisingly enough, few have treated the subject on the model of the standard text books of Pure and Applied Mathematics. One well recognized practice in these text books is to provide illustrative examples, easy exercises, which are straight deductions from the theory and which throw light on the various important points in the theory, exercises of moderate difficulty for the average student and more difficult exercises to test the ingenuity of the brightest student. Books on Statistics have so far tended to give either very difficult exercises taken from research papers or only very easy exercises and, in general, these exercises are too few to give the students enough practice and confidence. The present book is intended to fill the gap by providing a book on the same model as the books in calculus, differential equations, statics, dynamics, etc. and is based on our conviction that such a book is not only possible and desirable, but is also absolutely necessary. We have given, we believe, enough exercises of all types and these exercises and the text form an integrated pattern. A special feature is the large number of exercises in the bodies of the chapters, more than fifty per cent of which have been specially constructed to illustrate the theory and are so designed that after doing these, the student should not only have a better grasp of the theory, but should also know the motivation for the various steps. Another important feature is that unlike the other standard books on the subject, answers have been provided to almost all the exercises, so that the student can always decide for himself or herself, whether he or she has proceeded rightly or not.There is also a tendency in some recent books on Statistics, to slip over certain elementary, but basic topics, as well as to omit a large number of proofs, in order to cover as large a ground as possible in a small space. We have resisted this temptation, except in the last chapter on Testing of Hypotheses, Sequential Analysis and Distribution-free Methods. These topics are by their nature too wide to be covered completely in an introductory book, but we feel that a student of Statistics should be at least aware of the important basic concepts and results of these topics and, therefore, we have omitted a number of proofs in these* chapters. In all the earlier chapters, however, we have given proofs of almost all the statements that have been made and very often more than one proof have been given. Quite often alternative proofs have been sketched in the exercises that follow the main proof.While a great deal of emphasis has been placed on the Theory of Probability and its application in Mathematical Statistics, Descriptive Statistics and numerical problems on the same have also been given considerable space, specially in chapters on Measures of Location and Dispersion, Correlation and Regression, Fitting of Curves and Tests of Significance. Chapters 15 and 16 on Tests of Significance have been written, so as to be independent of chapters 13 and 14 on Exact Sampling Distributions, so that a student who is interested in these tests only and not on the derivations of the sampling distributions on which these tests are based can study these two chapters independently.As for mathematical pre-requisites, only the knowledge of calculus up to the B.A. (Pass) standard is assumed, so that the book can be useful for undergraduate students, who have Mathematics as one of their elective or subsidiary subjects. However, in view of the important part played by Beta and Gamma functions of Mathematical Statistics, their important properties have been discussed in Appendix I.Authors of a text book are always indebted to all the earlier books on the subject and as such we acknowledge with gratitude our debt to the authors and publishers of those books, which we have often consulted in the course of our teaching. We are deeply indebted to Professor R.A.Fisher and to Messrs Oliver and Boyd Ltd., Edinburgh for permission to reprint tables from Statistical Methods for Research Workers and to Prof.George W. Snedecor and Iowa State University Press, Iowa, U.S.A. for permission to reprint the tables from Statistical Methods (5th edition, 1956).In a book of this nature, a number of misprints and mistakes are likely to have crept in and we shall be grateful for any notice of the same as well as for any suggestions for improvement which will be gratefully acknowledged and given full consideration by us for the future.