This book is based on the lecture notes written for the advanced Ph.D. level
statistics courses delivered by the first author at the Wayne State Univer-
sity over the last decade. It has been easy to observe how the gap deepens
between applied (computational) and theoretical statistics. It has become
more diﬃcult to direct and mentor graduate students in the field of math-
ematical statistics. The research monographs in this field are extremely
diﬃcult to use as textbooks. Even in the best published lecture notes the
intensive material of original studies is typically included. On the other
hand, the classical courses in statistics that cover the traditional parametric
point and interval estimation methods and hypotheses testing are hardly
suﬃcient for the teaching goals in modern mathematical statistics.
In this book, we tried to give a general overview of the key statistical
topics, parametric and nonparametric, as a set of very special optimization
problems. As a criterion for optimality of estimators we chose minimax
risks, and we focused on asymptotically minimax rates of convergence for
large samples. Definitely, the selection of models presented in this book fol-
lows our preferences. Many very important problems and examples are not
included. The simplest models were deliberately selected for presentation,
and we consciously concentrated on the detailed proofs of all propositions.
We believe that mathematics students should be trained in proof-writing to
be better prepared for applications in statistics.
This textbook can form a reasonable basis for a two-semester course in
mathematical statistics. Every chapter is followed by a collection of exercises
consisting partly of verification of technical results, and partly of important