Expansions and Asymptotics for Statistics
Asymptotic methods provide important tools for approximating and analysing functions that arise in probability and statistics. Moreover, the conclusions of asymptotic analysis often supplement the conclusions obtained by numerical methods. Providing a broad toolkit of analytical methods, Expansions and Asymptotics for Statistics shows how asymptotics, when coupled with numerical methods, becomes a powerful way to acquire a deeper understanding of the techniques used in probability and statistics.
The book first discusses the role of expansions and asymptotics in statistics, the basic properties of power series and asymptotic series, and the study of rational approximations to functions. With a focus on asymptotic normality and asymptotic efficiency of standard estimators, it covers various applications, such as the use of the delta method for bias reduction, variance stabilisation, and the construction of normalising transformations, as well as the standard theory derived from the work of R.A. Fisher, H. Cramér, L. Le Cam, and others. The book then examines the close connection between saddle-point approximation and the Laplace method. The final chapter explores series convergence and the acceleration of that convergence.
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The contents about Laplace approximation and saddlepoint approximation are pretty good and deep enough, while assuming little mathematical background of the readers. Some topics like summation of series are not usual ones contained in other books. I learned a lot when I took Prof. Small's course using this book as our textbook. However, if you are looking for a comprehensive book for asymptotic statistics, covering topics like M-estimator, von Mise statitical functional, bootstrap, I would suggest van der vaart's asymptotic statistics. For thorough treatment on likelihood method and related higher order theory, I recommend Barndoff-Nielson and Cox's Inference and asymptotics.
This book is based on the lecture notes on asymptotic method in statistics given at University of Waterloo. It is easy to follow and also provides rich historical backgrounds on the development of theory. It can be a handy reference especially when you want to refresh or sharpen your knowledge of asymptotic techniques.
General series methods
Pade approximants and continued fractions
The delta method and its extensions
Optimality and likelihood asymptotics
The Laplace approximation and series
The saddlepoint method