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Mathematical Statistics with Resampling and R

ISBN: 978-1-118-02985-5
440 pages
September 2011, ©2011
Mathematical Statistics with Resampling and R (1118029852) cover image
This book bridges the latest software applications with the benefits of modern resampling techniques

Resampling helps students understand the meaning of sampling distributions, sampling variability, P-values, hypothesis tests, and confidence intervals. This groundbreaking book shows how to apply modern resampling techniques to mathematical statistics. Extensively class-tested to ensure an accessible presentation, Mathematical Statistics with Resampling and R utilizes the powerful and flexible computer language R to underscore the significance and benefits of modern resampling techniques.

The book begins by introducing permutation tests and bootstrap methods, motivating classical inference methods. Striking a balance between theory, computing, and applications, the authors explore additional topics such as:

  • Exploratory data analysis
  • Calculation of sampling distributions
  • The Central Limit Theorem
  • Monte Carlo sampling
  • Maximum likelihood estimation and properties of estimators
  • Confidence intervals and hypothesis tests
  • Regression
  • Bayesian methods

Throughout the book, case studies on diverse subjects such as flight delays, birth weights of babies, and telephone company repair times illustrate the relevance of the real-world applications of the discussed material. Key definitions and theorems of important probability distributions are collected at the end of the book, and a related website is also available, featuring additional material including data sets, R scripts, and helpful teaching hints.

Mathematical Statistics with Resampling and R is an excellent book for courses on mathematical statistics at the upper-undergraduate and graduate levels. It also serves as a valuable reference for applied statisticians working in the areas of business, economics, biostatistics, and public health who utilize resampling methods in their everyday work.

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Preface.

1 Data and Case Studies.

1.1 Case Study: Flight Delays.

1.2 Case Study: Birth Weights of Babies.

1.3 Case Study: Verizon Repair Times.

1.4 Sampling.

1.5 Parameters and Statistics.

1.6 Case Study: General Social Survey.

1.7 Sample Surveys.

1.8 Case Study: Beer and Hot Wings.

1.9 Case Study: Black Spruce Seedlings.

1.10 Studies.

1.11 Exercises.

2 Exploratory Data Analysis.

2.1 Basic Plots.

2.2 Numeric Summaries.

2.2.1 Center.

2.2.2 Spread.

2.2.3 Shape.

2.3 Boxplots.

2.4 Quantiles and Normal Quantile Plots.

2.5 Empirical Cumulative Distribution Functions.

2.6 Scatter Plots.

2.7 Skewness and Kurtosis.

2.8 Exercises.

3 Hypothesis Testing.

3.1 Introduction to Hypothesis Testing.

3.2 Hypotheses.

3.3 Permutation Tests.

3.3.1 Implementation Issues.

3.3.2 One-Sided and Two-Sided Tests.

3.3.3 Other Statistics.

3.3.4 Assumptions.

3.4 Contingency Tables.

3.4.1 Permutation Test for Independence.

3.4.2 Chi-Square Reference Distribution.

3.5 Chi-Square Test of Independence.

3.6 Test of Homogeneity.

3.7 Goodness-of-Fit: All Parameters Known.

3.8 Goodness-of-Fit: Some Parameters Estimated.

3.9 Exercises.

4 Sampling Distributions.

4.1 Sampling Distributions.

4.2 Calculating Sampling Distributions.

4.3 The Central Limit Theorem.

4.3.1 CLT for Binomial Data.

4.3.2 Continuity Correction for Discrete Random Variables.

4.3.3 Accuracy of the Central Limit Theorem.

4.3.4 CLT for Sampling Without Replacement.

4.4 Exercises.

5 The Bootstrap.

5.1 Introduction to the Bootstrap.

5.2 The Plug-In Principle.

5.2.1 Estimating the Population Distribution.

5.2.2 How Useful Is the Bootstrap Distribution?

5.3 Bootstrap Percentile Intervals.

5.4 Two Sample Bootstrap.

5.4.1 The Two Independent Populations Assumption.

5.5 Other Statistics.

5.6 Bias.

5.7 Monte Carlo Sampling: The "Second Bootstrap Principle".

5.8 Accuracy of Bootstrap Distributions.

5.8.1 Sample Mean: Large Sample Size.

5.8.2 Sample Mean: Small Sample Size.

5.8.3 Sample Median.

5.9 How Many Bootstrap Samples are Needed?

5.10 Exercises.

6 Estimation.

6.1 Maximum Likelihood Estimation.

6.1.1 Maximum Likelihood for Discrete Distributions.

6.1.2 Maximum Likelihood for Continuous Distributions.

6.1.3 Maximum Likelihood for Multiple Parameters.

6.2 Method of Moments.

6.3 Properties of Estimators.

6.3.1 Unbiasedness.

6.3.2 Efficiency.

6.3.3 Mean Square Error.

6.3.4 Consistency.

6.3.5 Transformation Invariance.

6.4 Exercises.

7 Classical Inference: Confidence Intervals.

7.1 Confidence Intervals for Means.

7.1.1 Confidence Intervals for a Mean, σ Known.

7.1.2 Confidence Intervals for a Mean, σ Unknown.

7.1.3 Confidence Intervals for a Difference in Means.

7.2 Confidence Intervals in General.

7.2.1 Location and Scale Parameters.

7.3 One-Sided Confidence Intervals.

7.4 Confidence Intervals for Proportions.

7.4.1 The Agresti–Coull Interval for a Proportion.

7.4.2 Confidence Interval for the Difference of Proportions.

7.5 Bootstrap t Confidence Intervals.

7.5.1 Comparing Bootstrap t and Formula t Confidence Intervals.

7.6 Exercises.

8 Classical Inference: Hypothesis Testing.

8.1 Hypothesis Tests for Means and Proportions.

8.1.1 One Population.

8.1.2 Comparing Two Populations.

8.2 Type I and Type II Errors.

8.2.1 Type I Errors.

8.2.2 Type II Errors and Power.

8.3 More on Testing.

8.3.1 On Significance.

8.3.2 Adjustments for Multiple Testing.

8.3.3 P-values Versus Critical Regions.

8.4 Likelihood Ratio Tests.

8.4.1 Simple Hypotheses and the Neyman–Pearson Lemma.

8.4.2 Generalized Likelihood Ratio Tests.

8.5 Exercises.

9 Regression.

9.1 Covariance.

9.2 Correlation.

9.3 Least-Squares Regression.

9.3.1 Regression Toward the Mean.

9.3.2 Variation.

9.3.3 Diagnostics.

9.3.4 Multiple Regression.

9.4 The Simple Linear Model.

9.4.1 Inference for α and β.

9.4.2 Inference for the Response.

9.4.3 Comments About Assumptions for the Linear Model.

9.5 Resampling Correlation and Regression.

9.5.1 Permutation Tests.

9.5.2 Bootstrap Case Study: Bushmeat.

9.6 Logistic Regression.

9.6.1 Inference for Logistic Regression.

9.7 Exercises.

10 Bayesian Methods.

10.1 Bayes’ Theorem.

10.2 Binomial Data, Discrete Prior Distributions.

10.3 Binomial Data, Continuous Prior Distributions.

10.4 Continuous Data.

10.5 Sequential Data.

10.6 Exercises.

11 Additional Topics.

11.1 Smoothed Bootstrap.

11.1.1 Kernel Density Estimate.

11.2 Parametric Bootstrap.

11.3 The Delta Method.

11.4 Stratified Sampling.

11.5 Computational Issues in Bayesian Analysis.

11.6 Monte Carlo Integration.

11.7 Importance Sampling.

11.7.1 Ratio Estimate for Importance Sampling.

11.7.2 Importance Sampling in Bayesian Applications.

11.8 Exercises.

Appendix A Review of Probability.

A.1 Basic Probability.

A.2 Mean and Variance.

A.3 The Mean of a Sample of Random Variables.

A.4 The Law of Averages.

A.5 The Normal Distribution.

A.6 Sums of Normal Random Variables.

A.7 Higher Moments and the Moment Generating Function.

Appendix B Probability Distributions.

B.1 The Bernoulli and Binomial Distributions.

B.2 The Multinomial Distribution.

B.3 The Geometric Distribution.

B.4 The Negative Binomial Distribution.

B.5 The Hypergeometric Distribution.

B.6 The Poisson Distribution.

B.7 The Uniform Distribution.

B.8 The Exponential Distribution.

B.9 The Gamma Distribution.

B.10 The Chi-Square Distribution.

B.11 The Student's t Distribution.

B.12 The Beta Distribution.

B.13 The F Distribution.

B.14 Exercises.

Appendix C Distributions Quick Reference.

Solutions to Odd-Numbered Exercises.

Bibliography.

Index.

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LAURA CHIHARA, PhD, is Professor of Mathematics at Carleton College. She has extensive experience teaching mathematical statistics and applied regression analysis. She has supervised undergraduates working on statistics projects for local businesses and organizations such as Target Corporation and the Minnesota Pollution Control Agency. Dr. Chihara has experience with S+ and R from her work at Insightful Corporation (formerly MathSoft) and in statistical consulting.

TIM HESTERBERG, PhD, is Senior Ads Quality Statistician at Google. He was a senior research scientist for Insightful Corporation and led the development of S+Resample and other S+ and R software. Dr. Hesterberg has published numerous articles in the areas of bootstrap and related resampling techniques, Monte Carlo simulation methodology, modern regression, tectonic deformation estimation, and electric demand forecasting.

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"Mathematical Statistics with Resampling and R is a great resource for intermediate and advanced statistics students who want to achieve an indepth understanding of resampling techniques backed by practical implementation." (Book Pleasures, 2012)

"It is highly recommended to someone with a good background in mathematics, probability, and basic statistics who wants to learn about the theory and about resampling and how it relates to traditional methods, and how to implement resamplinjg in R. The book is also a wonderful source of simulations to support the teaching of statistics." (Journal of Biopharmaceutical Statistics, 2011)

"It is less demanding mathematically, more applied in its emphasis, and more modern in content than the usual book, which makes it a good choice if you want a modern applied book at the level of Larsen and Marx (1986)."- George W. Cobb, Mount Holyoke College Department of Mathematics and Statsitics (Chilean Journal of Statistics, 1 April 2011)

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