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Mathematical Statistics with Resampling and R

Book Description

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.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface
  7. 1 Data and Case Studies
    1. 1.1 Case Study: Flight Delays
    2. 1.2 Case Study: Birth Weights of Babies
    3. 1.3 Case Study: Verizon Repair Times
    4. 1.4 Sampling
    5. 1.5 Parameters and Statistics
    6. 1.6 Case Study: General Social Survey
    7. 1.7 Sample Surveys
    8. 1.8 Case Study: Beer and Hot Wings
    9. 1.9 Case Study: Black Spruce Seedlings
    10. 1.10 Studies
    11. 1.11 Exercises
  8. 2 Exploratory Data Analysis
    1. 2.1 Basic Plots
    2. 2.2 Numeric Summaries
    3. 2.3 Boxplots
    4. 2.4 Quantiles and Normal Quantile Plots
    5. 2.5 Empirical Cumulative Distribution Functions
    6. 2.6 Scatter Plots
    7. 2.7 Skewness and Kurtosis
    8. 2.8 Exercises
  9. 3 Hypothesis Testing
    1. 3.1 Introduction to Hypothesis Testing
    2. 3.2 Hypotheses
    3. 3.3 Permutation Tests
    4. 3.4 Contingency Tables
    5. 3.5 Chi-Square Test of Independence
    6. 3.6 Test of Homogeneity
    7. 3.7 Goodness-of-Fit: All Parameters Known
    8. 3.8 Goodness-of-Fit: Some Parameters Estimated
    9. 3.9 Exercises
  10. 4 Sampling Distributions
    1. 4.1 Sampling Distributions
    2. 4.2 Calculating Sampling Distributions
    3. 4.3 The Central Limit Theorem
    4. 4.4 Exercises
  11. 5 The Bootstrap
    1. 5.1 Introduction to the Bootstrap
    2. 5.2 The Plug-In Principle
    3. 5.3 Bootstrap Percentile Intervals
    4. 5.4 Two Sample Bootstrap
    5. 5.5 Other Statistics
    6. 5.6 Bias
    7. 5.7 Monte Carlo Sampling: The “Second Bootstrap Principle”
    8. 5.8 Accuracy of Bootstrap Distributions
    9. 5.9 How Many Bootstrap Samples are Needed?
    10. 5.10 Exercises
  12. 6 Estimation
    1. 6.1 Maximum Likelihood Estimation
    2. 6.2 Method of Moments
    3. 6.3 Properties of Estimators
    4. 6.4 Exercises
  13. 7 Classical Inference: Confidence Intervals
    1. 7.1 Confidence Intervals for Means
    2. 7.2 Confidence Intervals in General
    3. 7.3 One-Sided Confidence Intervals
    4. 7.4 Confidence Intervals for Proportions
    5. 7.5 Bootstrap t Confidence Intervals
    6. 7.6 Exercises
  14. 8 Classical Inference: Hypothesis Testing
    1. 8.1 Hypothesis Tests for Means and Proportions
    2. 8.2 Type I and Type II Errors
    3. 8.3 More on Testing
    4. 8.4 Likelihood Ratio Tests
    5. 8.5 Exercises
  15. 9 Regression
    1. 9.1 Covariance
    2. 9.2 Correlation
    3. 9.3 Least-Squares Regression
    4. 9.4 The Simple Linear Model
    5. 9.5 Resampling Correlation and Regression
    6. 9.6 Logistic Regression
    7. 9.7 Exercises
  16. 10 Bayesian Methods
    1. 10.1 Bayes’ Theorem
    2. 10.2 Binomial Data, Discrete Prior Distributions
    3. 10.3 Binomial Data, Continuous Prior Distributions
    4. 10.4 Continuous Data
    5. 10.5 Sequential Data
    6. 10.6 Exercises
  17. 11 Additional Topics
    1. 11.1 Smoothed Bootstrap
    2. 11.2 Parametric Bootstrap
    3. 11.3 The Delta Method
    4. 11.4 Stratified Sampling
    5. 11.5 Computational Issues in Bayesian Analysis
    6. 11.6 Monte Carlo Integration
    7. 11.7 Importance Sampling
    8. 11.8 Exercises
  18. Appendix A Review of Probability
    1. A.1 Basic Probability
    2. A.2 Mean and Variance
    3. A.3 The Mean of a Sample of Random Variables
    4. A.4 The Law of Averages
    5. A.5 The Normal Distribution
    6. A.6 Sums of Normal Random Variables
    7. A.7 Higher Moments and the Moment Generating Function
  19. Appendix B Probability Distributions
    1. B.1 The Bernoulli and Binomial Distributions
    2. B.2 The Multinomial Distribution
    3. B.3 The Geometric Distribution
    4. B.4 The Negative Binomial Distribution
    5. B.5 The Hypergeometric Distribution
    6. B.6 The Poisson Distribution
    7. B.7 The Uniform Distribution
    8. B.8 The Exponential Distribution
    9. B.9 The Gamma Distribution
    10. B.10 The Chi-Square Distribution
    11. B.11 The Student’s t Distribution
    12. B.12 The Beta Distribution
    13. B.13 The F Distribution
    14. B.14 Exercises
  20. Appendix C Distributions Quick Reference
  21. Solutions to Odd-Numbered Exercises
  22. Bibliography
  23. Index