A Course in Mathematical Statistics and Large Sample Theory

This is a very nice book suitable for a theoretical statistics course after having worked through something at the level of Casella & Berger, as well as some measure theory. It consists of three parts.

The first part is basic (non-asymptotic) statistics at a fairly sophisticated level. It includes chapters on Decision Theory, Methods of Estimation, Sufficient Statistics & Exponential Families, and Hypothesis Testing. Earlier chapters have a few exercises, that later chapters have about a dozen with hints/answers to many in the back of the book.

The second part on asymptotic theory has chapters on Consistency and Asymptotic Distributions, Large Sample Theory of Estimation in Parametric Models, Tests, the Nonparametric Bootstrap, and Nonparametric Curve Estimation. The sophistication level ramps up, and here a previous course in measure theoretic probability is needed. For example, a nice two page informal introduction to Brownian motion is provided — but more familiarity is beneficial. In the first part of this section there are a couple dozen exercises for each chapter, but that thins out in the later chapters, and neither hints nor solutions are provided for this or the third part.

The third part consists of several short chapters on interesting topics: Edgeworth Expansions and the Bootstrap, Fréchet Means and Nonparametric Inference on Non-Euclidean Geometric Spaces, Multiple Testing and the False Discovery Rate, Markov Chain Monte Carlo (MCMC) Simulation and Bayes, and a Miscellaneous Topics chapter than includes Classification/Machine Learning, Principal Component Analysis, and the Sequential Probability Ratio Test. These chapters are more suitable for a topics course, as they are very brief and only get the reader started on the areas. There are only a few exercises in the entire last section.

In addition to the exercises, which range from doable to interesting, there are several projects scattered throughout the text. The explanations are clear and crisp, and the presentation is interesting. I have not found any typos that affect understanding.

If you are teaching and advanced theoretical l statistics course have a look at the Table of Contents and preface at Springer’s web site. Even if you aren’t, the book would be a worthy addition to your statistics library.

Peter Rabinovitch is a Senior Performance Engineer at Akamai, and as been doing data science since long before “data science” was a thing.