Mathematical Statistics: Exercises and Solutions
This book is a companion to the author’s textbook ‘Mathematical Statistics’ (2nd ed., Springer, 2003), which contains over 900 exercises. This collection consists of 400 exercises and their solutions. Most of the exercises (over 95%) are introduced in the cited textbook. The reader should have a good knowledge of advanced calculus, real analysis and measure theory. The book is divided into seven chapters: 1. Probability theory (measure and integral, distribution functions, random variables), 2. Fundamentals of statistics (sufficiency, risk functions, admissibility, consistency, Bayes rule), 3. Unbiased estimation (uniformly minimum variance unbiased estimators, Fisher information, U-statistics, linear models), 4. Estimation in parametric models (conjugate priors, posterior distributions, minimum risk invariant estimators, least squares estimators, maximum likelihood estimators, asymptotic relative efficiency), 5. Estimation in nonparametric models (Mallows’ distance, influence function, L-functionals, Hodges-Lehmann estimator), 6. Hypothesis test (uniformly most powerful tests, likelihood ratio tests), 7. Confidence sets (Fieller’s confidence sets, pivotal quantity, uniformly most accurate confidence sets).
The collection is a stand-alone book. It is written very rigorously and solutions are presented in detail. It can be recommended as a source of solved problems for teachers and students of advanced mathematical statistics.
I will finish with two remarks. First, I think that it may be of some interest to reproduce a very simple exercise (Ex. 9, p. 7): Let F be a cumulative distribution function on the real line and a a real number. Show that . Second, the reviewer would like to point out a misprint to prove that he studied the book: the expression mk.