Page 12 - Probability and Statistical Inference

Preface

This textbook aims to foster the theory of both probability and statistical
inference for first-year graduate students in statistics or other areas in which
a good understanding of statistical concepts is essential. It can also be used
as a textbook in a junior/senior level course for statistics or mathematics/
statistics majors, with emphasis on concepts and examples. The book includes
the core materials that are usually taught in a two-semester or three-quarter
sequence.
A distinctive feature of this book is its set of examples and exercises.
These are essential ingredients in the total learning process. I have tried to
make the subject come alive through many examples and exercises.
This book can also be immensely helpful as a supplementary text in a
significantly higher level course (for example, Decision Theory and Advanced
Statistical Inference) designed for second or third year graduate students in
statistics.
The prerequisite is one year’s worth of calculus. That should be enough
to understand a major portion of the book. There are sections for which
some familiarity with linear algebra, multiple integration and partial
differentiation will be beneficial. I have reviewed some of the important
mathematical results in Section 1.6.3. Also, Section 4.8 provides a selected
review of matrices and vectors.
The first four chapters introduce the basic concepts and techniques in
probability theory, including the calculus of probability, conditional probability,
independence of events, Bayes’s Theorem, random variables, probability
distributions, moments and moment generating functions (mgf), probability
generating functions (pgf), multivariate random variables, independence of
random variables, standard probability inequalities, the exponential family
of distributions, transformations and sampling distributions. Multivariate
normal, t and F distributions have also been briefly discussed. Chapter 5
develops the notions of convergence in probability, convergence in distribution,
the central limit theorem (CLT) for both the sample mean and sample
variance, and the convergence of the density functions of the Chi-square, t
and F distributions.
The remainder of the book systematically develops the concepts of
statistical inference. It is my belief that the concept of “sufficiency” is the
heart of statistical inference and hence this topic deserves appropriate care
and respect in its treatment. I introduce the fundamental notions of sufficiency,
Neyman factorization, information, minimal sufficiency, completeness, and
ancillarity very early, in Chapter 6. Here, Basu’s Theorem and the location,
v