This book originally grew out of my notes on the statistical inference courses at the
Department of Statistics, University of Mumbai. I have experienced that reasonably
good M.Sc. (Statistics) students are many a time not able to understand or solve
problems from some available texts on statistical inference. These books are
excellent in terms of content, but the presentation is highly sophisticated. For
instance, proofs of various theorems are given in brief and a few examples are
provided. To overcome this difficulty, I have solved many examples and, wherever
necessary, a program in R is also given. Further, important proofs in this book are
presented in such a manner that they are easy to understand.
Through this book, we expect students to know matrix algebra, calculus,
probability theory, and distribution theory. This book will serve as an excellent tool
for teaching statistical inference courses. The book consists of many solved and
unsolved problems. Instructors can assign homework problems from the exercises
and students will find the solved examples hugely beneficial in solving the exercise
problems.
In “Prerequisite”, we have discussed some basic concepts like distribution
function and order statistics and illustrated them by using interesting examples.
Chapter 1 deals with sufficiency and completeness. In this chapter, we have solved
37 examples. Chapter 2 deals with unbiased estimation. In the last 30 years of my
teaching, I found that students were always confused about the relationship between
sufficiency and unbiasedness. We have explained this relationship with various
examples in this chapter. Chapter 3 is devoted to method of moments and maximum
likelihood. In Chap. 4, we deal with lower bound for the variance of an unbiased
estimator. Popular concepts like Cramer–Rao (1945, 1946) and Bhattacharya
(1946, 1950) lower bound are discussed in detail. Chapter 4 also deals with
Chapman and Robbins (1951) and Kiefer (1952) lower bound for the variance of an
estimate but does not require regularity conditions. In Chap. 5, the concept of
consistency is discussed in detail and illustrated by using different examples. In
Chap. 6, Bayesian estimation is briefly discussed. Chapters 7 and 8 are significantly
large chapters. Testing of hypothesis is studied in Chap. 7, whereas unbiased and
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other tests are studied in Chap. 8. We have given R programs in various chapters.
No originality is claimed except perhaps in the presentation of the material.
It will prove difficult to thank all my friends who have contributed in some or
other way to make this book a reality. I am thankful to Prof. R.B. Bapat for his
valuable suggestions to improve upon the content and presentation of the book.
I also thank Dr. T.V. Ramanathan for making some valuable suggestions. I am
thankful to Shamim Ahmad, senior editor at Springer India for encouraging me to
publish this book through Springer and making it easy to go through the process.
I thank Prof. Seema C. for reading the book for language. I am equally thankful to
Dr. Alok Dabade, Prof. Shailaja Kelkar, Dr. Mehdi Jabbari Nooghabi, Prof.
S. Annapurna and Prof. Mandar Bhanushe for various academic discussions related
to the book and drawing figures. I am also very thankful to my son Anand and
daughter Vaidehi who helped me solve various problems. Further, I am thankful to
my wife Dr. (Mrs.) Vaijayanti for the insightful discussions on our book.
We are grateful to Prof. Y.S. Sathe and Late Prof. M.N. Vartak for the diverse
discussions which were helpful in understanding statistical inference. These discussions were particularly helpful in solving problems on UMVUE and testing
of hypotheses. We are thankful to Prof. B.V. Dhandra, Dr. D.B. Jadhav, and
Prof. D.T. Jadhav for providing their M.Phil. dissertations.
In spite of my best efforts, there might be some errors and misprints in the
presentation. I owe these mistakes and request the readers to kindly bring them to
my notice.
Ulhas Jayram Dixi