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This book began as a revision of Elements of Computational Statistics, published by Springer in 2002. That book covered computationally-intensive statistical methods from the perspective of statistical applications, rather than from the standpoint of statistical computing.
Most of the students in my courses in computational statistics were in a program that required multiple graduate courses in numerical analysis, and so in my course in computational statistics, I rarely covered topics in numerical linear algebra or numerical optimization, for example. Over the years, however, I included more discussion of numerical analysis in my computational statistics courses. Also over the years I have taught numerical methods courses with no or very little statistical content. I have also accumulated a number of corrections and small additions to the elements of computational statistics. The present book includes most of the topics from Elements and also incorporates this additional material. The emphasis is still on computationally intensive statistical methods, but there is a substantial portion on the numerical methods supporting the statistical applications.
I have attempted to provide a broad coverage of the field of computational statistics. This obviously comes at the price of depth.
Part I, consisting of one rather long chapter, presents some of the most important concepts and facts over a wide range of topics in intermediate-level mathematics, probability and statistics, so that when I refer to these concepts in later parts of the book, the reader has a frame of reference.
Part I attempts to convey the attitude that computational inference, together with exact inference and asymptotic inference, is an important component of statistical methods.
Many statements in Part I are made without any supporting argument, but references and notes are given at the end of the chapter. Most readers and students in courses in statistical computing or computational statistics will be familiar with a substantial proportion of the material in Part I, but I do not recommend skipping the chapter. If readers are already familiar with the material, they should just read faster. The perspective in this chapter is that of the big picture. As is often apparent in oral exams, many otherwise good students lack a basic understanding of what it is all about.
A danger is that the student or the instructor using the book as a text will too quickly gloss over Chapter 1 and miss some subtle points.
Part II addresses statistical computing, a topic dear to my heart. There are many details of the computations that can be ignored by most statisticians, but no matter at what level of detail a statistician needs to be familiar with the computational topics of Part II, there are two simple, higher-level facts that all statisticians should be aware of and which I state often in this book:
Computer numbers are not the same as real numbers, and the arithmetic operations on computer numbers are not exactly the same as those of ordinary arithmetic.
and
The form of a mathematical expression and the way the expression should be evaluated in actual practice may be quite different.
Regarding the first statement, some of the differences in real numbers and computer numbers are summarized in Table 2.1 on page 98.
A prime example of the second statement is the use of the normal equations in linear regression, XTXb = XTy. It is quite appropriate to write and discuss these equations. We might consider the elements of XTX, and we might even write the least squares estimate of β as b = (XTX) -1XTy. That does not mean that we ever actually compute XTX or (XTX) -1, although we may compute functions of those matrices or even certain elements of them.
Preliminaries
Mathematical and Statistical Preliminaries
Statistical Computing
Computer Storage and Arithmetic
Algorithms and Programming
Approximation of Functions and Numerical Quadrature
Numerical Linear Algebra
Solution of Nonlinear Equations and Optimization
Generation of Random Numbers
Methods of Computational Statistics
Graphical Methods in Computational Statistics
Tools for Identification of Structure in Data
Estimation of Functions
Monte Carlo Methods for Statistical Inference
Data Randomization, Partitioning, and Augmentation
Bootstrap Methods
Exploring Data Density and Relationships
Estimation of Probability Density Functions Using Parametric Models
Parametric Families
Nonparametric Estimation of Probability Density Functions
Statistical Learning and Data Mining
Statistical Models of Dependencies
Appendices
Monte Carlo Studies in Statistics
Some Important Probability Distributions
Notation and Definitions
Solutions and Hints for Selected Exercise