Statistical Computing with R, Second Edition

Maria L. Rizzo

• Publisher: Chapman and Hall/CRC
• Year: 2019
• Print ISBN: 9781466553323
• eBook ISBN: 9780429527760
• Topic: Economics
• Subtopic: Probability & Statistics
• Index: Economics

Computational statistics and statistical computing are two areas that employ computational, graphical, and numerical approaches to solve statistical problems, making the versatile R language an ideal computing environment for these fields. This second edition continues to encompass the traditional core material of computational statistics, with an

Chapter 1

Introduction

1.1 Statistical Computing

Computational statistics and statistical computing are two areas within statistics that may be broadly described as computational, graphical, and numerical approaches to solving statistical problems. Statistical computing traditionally has more emphasis on numerical methods and algorithms, such as optimization and random number generation, while computational statistics may encompass such topics as exploratory data analysis, Monte Carlo methods, and data partitioning, etc. However, most researchers who apply computationally intensive methods in statistics use both computational statistics and statistical computing methods; there is much overlap and the terms are used differently in different contexts and disciplines. Gentle [118] and Givens and Hoeting [129] use “computational statistics” to encompass all the relevant topics that should be covered in a modern introductory text, so that “statistical computing” is somewhat absorbed under this more broad definition of computational statistics. On the other hand, journals and professional organizations seem to use both terms to cover similar areas.

This book encompasses parts of both of these subjects, because a first course in computational methods for statistics necessarily includes both. Some examples of topics covered are described below.

Monte Carlo methods refer to a diverse collection of methods in statistical inference and numerical analysis where simulation is used. Many statistical problems can be approached through some form of Monte Carlo integration. In parametric bootstrap, samples are generated from a given probability distribution to compute probabilities, gain information about sampling distributions of statistics such as bias and standard error, to assess the performance of procedures in statistical inference, and to compare the performance of competing methods for the same problem. Resampling methods such as the ordinary bootstrap and jackknife are nonparametric methods that can be applied when the distribution of the random variable or a method to simulate it directly is unavailable. The need for Monte Carlo analysis also arises because in many problems, an asymptotic approximation is unsatisfactory or intractable. The convergence to the limit distribution may be too slow, or we require results for finite samples; or the asymptotic distribution has unknown parameters. Monte Carlo methods are covered in Chapters 6–11. The first tool needed in a simulation is a method for generating psuedo-random samples; these methods are covered in Chapters 3 and 4.

Markov Chain Monte Carlo (MCMC) methods are based on an algorithm to sample from a specified target probability distribution that is the stationary distribution of a Markov chain. These methods are widely applied for problems arising in Bayesian analysis, and in such diverse fields as computational physics and computational finance. Markov Chain Monte Carlo methods are covered in Chapter 11.

Several special topics also deserve an introduction in a survey of computationally intensive methods. Density estimation (Chapter 12) provides a nonparametric estimate of a density, which has many applications in addition to estimation, ranging from exploratory data analysis to cluster analysis. Computational methods are essential for the visualization of multivariate data and reduction of dimensionality. The increasing interest in massive and streaming data sets, and high dimensional data arising in applications of biology and engineering, for example, demand improved and new computational approaches for multivariate analysis and visualization. Chapter 5 is an introduction to methods for visualization of multivariate data. A review of selected topics in numerical methods such as root finding and numerical integration is presented in Chapter 13. An introduction to optimization using R is covered in Chapter 14.

A final chapter of optional material specific to R programming should be accessible to readers after covering Chapter 3. Programming topics such as benchmarking, efficiency and code profiling are covered in Chapter 15. Several years ago with the release of Rcpp [82, 83], writing R extensions in compiled libraries became much simpler so that most experienced R users with a modest amount of background in C++ can easily integrate compiled C++ functions with R code. Some simple examples are illustrated in the final chapter of the book for those users who are interested.

Many references can be recommended for further reading on these topics. Efron and Hastie [89] provide an up-to-date review of how modern statistics has evolved in the computer age. Gentle [118, 119] and the volume edited by Gentle, et al. [120] have thorough coverage of topics in computational statistics. A survey of methods in statistical computing is covered in Kundu and Basu [170]. Givens and Hoeting [129] is a recent graduate text on computational statistics and statistical computing. Hardle et al. [139] is an introductory text with examples in R. Martinez and Martinez [197] is an accessible introduction to computational statistics, with numerous examples in MATLAB^®. Books that primarily cover Monte Carlo methods or resampling methods include Davison and Hinkley [68], Efron and Tibshirani [91], Hjorth [149], Liu [186], Chernick [50] and Robert and Casella [240]. Statistical learning is a closely related topic that applies computational methods to solve a wide range of problems in modern statistics; see Hastie et al. [143] and James et al. [157]. On density estimation see Scott [264] and Silverman [268]. A good resource for applied linear models in R ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface to the Second Edition
• Preface to the First Edition
• 1 Introduction
• 2 Probability and Statistics Review
• 3 Methods for Generating Random Variables
• 4 Generating Random Processes
• 5 Visualization of Multivariate Data
• 6 Monte Carlo Integration and Variance Reduction
• 7 Monte Carlo Methods in Inference
• 8 Bootstrap and Jackknife
• 9 Resampling Applications
• 10 Permutation Tests
• 11 Markov Chain Monte Carlo Methods
• 12 Probability Density Estimation
• 13 Introduction to Numerical Methods in R
• 14 Optimization
• 15 Programming Topics
• Notation
• Bibliography
• Index