Statistical Computing with R 2nd Edition by Maria Rizzo ISBN 0429527764 9780429527760

1 Introduction

1.1 Statistical Computing

1.2 The R Environment

1.3 Getting Started with R and RStudio

1.4 Basic Syntax

1.5 Using the R Online Help System

1.6 Distributions and Statistical Tests

1.7 Functions

1.8 Arrays, Data Frames, and Lists

1.9 Formula Specification

1.10 Graphics

1.11 Introduction to ggplot

1.12 Workspace and Files

1.12.1 The Working Directory

1.12.2 Reading Data from External Files

1.12.3 Importing/Exporting .csv Files

1.13 Using Scripts

1.14 Using Packages

1.15 Using R Markdown and knitr

2 Probability and Statistics Review

2.1 Random Variables and Probability

2.2 Some Discrete Distributions

2.3 Some Continuous Distributions

2.4 Multivariate Normal Distribution

2.5 Limit Theorems

2.6 Statistics

2.7 Bayes’ Theorem and Bayesian Statistics

2.8 Markov Chains

3 Methods for Generating Random Variables

3.1 Introduction

3.2 The Inverse Transform Method

3.2.1 Inverse Transform Method, Continuous Case

3.2.2 Inverse Transform Method, Discrete Case

3.3 The Acceptance-Rejection Method

3.4 Transformation Methods

3.5 Sums and Mixtures

3.6 Multivariate Distributions

3.6.1 Multivariate Normal Distribution

3.6.2 Mixtures of Multivariate Normals

3.6.3 Wishart Distribution

3.6.4 Uniform Distribution on the d-Sphere

Exercises

4 Generating Random Processes

4.1 Stochastic Processes

4.1.1 Poisson Processes

4.1.2 Renewal Processes

4.1.3 Symmetric Random Walk

4.2 Brownian Motion

Exercises

5 Visualization of Multivariate Data

5.1 Introduction

5.2 Panel Displays

5.3 Correlation Plots

5.4 Surface Plots and 3D Scatter Plots

5.4.1 Surface Plots

5.4.2 Three-dimensional Scatterplot

5.5 Contour Plots

5.6 Other 2D Representations of Data

5.6.1 Andrews Curves

5.6.2 Parallel Coordinate Plots

5.6.3 Segments, Stars, and Other Representations

5.7 Principal Components Analysis

5.8 Other Approaches to Data Visualization

5.9 Additional Resources

Exercises

6 Monte Carlo Integration and Variance Reduction

6.1 Introduction

6.2 Monte Carlo Integration

6.2.1 Simple Monte Carlo Estimator

6.2.2 Variance and Efficiency

6.3 Variance Reduction

6.4 Antithetic Variables

6.5 Control Variates

6.5.1 Antithetic Variate as Control Variate

6.5.2 Several Control Variates

6.5.3 Control Variates and Regression

6.6 Importance Sampling

6.7 Stratified Sampling

6.8 Stratified Importance Sampling

Exercises

R Code

7 Monte Carlo Methods in Inference

7.1 Introduction

7.2 Monte Carlo Methods for Estimation

7.2.1 Monte Carlo Estimation and Standard Error

7.2.2 Estimation of MSE

7.2.3 Estimating a Confidence Level

7.3 Monte Carlo Methods for Hypothesis Tests

7.3.1 Empirical Type I Error Rate

7.3.2 Power of a Test

7.3.3 Power Comparisons

7.4 Application: “Count Five” Test for Equal Variance

Exercises

8 Bootstrap and Jackknife

8.1 The Bootstrap

8.1.1 Bootstrap Estimation of Standard Error

8.1.2 Bootstrap Estimation of Bias

8.2 The Jackknife

8.3 Bootstrap Confidence Intervals

8.3.1 The Standard Normal Bootstrap Confidence Interval

8.3.2 The Basic Bootstrap Confidence Interval

8.3.3 The Percentile Bootstrap Confidence Interval

8.3.4 The Bootstrap t Interval

8.4 Better Bootstrap Confidence Intervals

8.5 Application: Cross Validation

Exercises

9 Resampling Applications

9.1 Jackknife-after-Bootstrap

9.2 Resampling for Regression Models

9.2.1 Resampling Cases

9.2.2 Resampling Errors (Model Based)

9.3 Influence

9.3.1 Empirical Influence Values for a Statistic

9.3.2 Jackknife-after-Bootstrap Plots

Exercises

10 Permutation Tests

10.1 Introduction

10.2 Tests for Equal Distributions

10.3 Multivariate Tests for Equal Distributions

10.3.1 Nearest Neighbor Tests

10.3.2 Energy Test for Equal Distributions

10.4 Application: Distance Correlation

Exercises

11 Markov Chain Monte Carlo Methods

11.1 Introduction

11.1.1 Integration Problems in Bayesian Inference

11.1.2 Markov Chain Monte Carlo Integration

11.2 The Metropolis-Hastings Algorithm

11.2.1 Metropolis-Hastings Sampler

11.2.2 The Metropolis Sampler

11.2.3 Random Walk Metropolis

11.2.4 The Independence Sampler

11.3 The Gibbs Sampler

11.4 Monitoring Convergence

11.4.1 Why Monitor Convergence

11.4.2 Methods for Monitoring Convergence

11.4.3 The Gelman-Rubin Method

11.5 Application: Change Point Analysis

Exercises

R Code

12 Probability Density Estimation

12.1 Univariate Density Estimation

12.1.1 Histograms

12.1.2 Frequency Polygon Density Estimate

12.1.3 The Averaged Shifted Histogram

12.2 Kernel Density Estimation

12.3 Bivariate and Multivariate Density Estimation

12.3.1 Bivariate Frequency Polygon

12.3.2 Bivariate ASH

12.3.3 Multidimensional Kernel Methods

12.4 Other Methods of Density Estimation

Exercises

R Code

13 Introduction to Numerical Methods in R

13.1 Introduction

13.2 Root-finding in One Dimension

13.3 Numerical Integration

13.4 Maximum Likelihood Problems

13.5 Application: Evaluating an Expected Value

Exercises

14 Optimization

14.1 Introduction

14.2 One-dimensional Optimization

14.3 Maximum Likelihood Estimation with mle

14.4 Two-dimensional Optimization

14.5 The EM Algorithm

14.6 Linear Programming – The Simplex Method

14.7 Application: Game Theory

Exercises

15 Programming Topics

15.1 Introduction

15.2 Benchmarking: Comparing the Execution Time of Code

15.2.1 Using the microbenchmark Package

15.2.2 Using the rbenchmark Package

15.3 Profiling

15.4 Object Size, Attributes, and Equality

15.4.1 Object Size

15.4.2 Attributes of Objects

15.4.3 Comparing Objects for Equality

15.5 Finding Source Code

15.5.1 Finding R Function Code

15.5.2 Methods

15.5.3 Methods and Functions in Packages

15.5.4 Compiled Code

15.6 Linking C/C++ Code Using Rcpp

15.7 Application: Baseball Data

Exercises

Notation

Bibliography

Index