An Introduction to Mathematical Statistics and Its Applications (Fourth Edition)

Preface for the 4th edition

We are pleased that our text has been sufficiently well received to justify this fourth edition. Students and instructors who use the text like the coupling of the rigorous and structured treatment of probability and statistics with real-world case studies and examples. The users of the book have been helpful in pointing out ways to improve our prese...

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Preface for the 4th edition

We are pleased that our text has been sufficiently well received to justify this fourth edition. Students and instructors who use the text like the coupling of the rigorous and structured treatment of probability and statistics with real-world case studies and examples. The users of the book have been helpful in pointing out ways to improve our presentation. The changes found in this fourth edition reflect the many helpful suggestions we have received, as well as our own experience in teaching from the text.

Our first goal in writing this fourth edition was to continue strengthening the bridge between theory and practice. To that end, we have added sections at the end of each chapter called Taking a Second Look at Statistics. These sections discuss practical problems in applying the ideas in the chapter and also deal with common misunderstandings or faulty approaches. We also have included a new section on Bayesian estimation that integrates well into Chapter 5 on estimation and gives another view of how estimation can be applied. It introduces students to Bayesian ideas and also serves to reinforce the main concepts of estimation.

Some ideas that are useful and important lie beyond the mathematical scope of the text. To explore such topics within the mathematical context of the book, we have increased and enhanced the material on simulation and on the use of Monte Carlo studies. Since MINITAB is the main tool for simulations and demonstrating computer computations, the MINITAB sections have been rewritten to conform to Version 14, the latest release.

A barrier to efficient coverage of the book has been the length of time required to cover Chapters 2 and 3. One of the major changes in the fourth edition is a substantial revision of basic probability material. Chapters 2 and 3 have been reorganized and rewritten with the goal of a streamlined presentation. These chapters are now easier to teach and can be covered in less time, yet without loss of rigor.

In that same spirit, we have also improved and streamlined the development of the t, chi square and F distributions in Chapter 7, the heart of the book. The material there has been rewritten to simplify the development of the chi square distribution. In addition, we have made a much better division between the theoretical results and their applications.

Because of the efficiencies in the new edition, covering Chapters 1-7 plus other additional topics in one semester is now possible.

All in all, we feel that this new edition furthers our objective of writing a book that emphasizes the interrelation between probability theory, mathematical statistics, and data analysis. As in previous editions, real-world case studies and historical anecdotes provide valuable tools to effect the integration of these three areas. Our experience in the classroom has strengthened our belief in this approach. Students can better grasp the importance of each area when seen in the context of the other two.

Preface
1 Introduction
1.1 A Brief History
1.2 Some Examples
1.3 A Chapter Summary
2 Probability
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Preface
1 Introduction
1.1 A Brief History
1.2 Some Examples
1.3 A Chapter Summary
2 Probability
2.1 Introduction...............
2.2 Sample Spaces and the Algebra of Sets
2.3 . The Probability Function .
2.4 Conditional Probability .
2.5 Independence . . . . . . .
2.6 Combinatorics.......
2.7 Combinatorial Probability
2.8 Taking a Second Look at Statistics (Enumeration and
Monte Carlo Techniques) ................
3 Random Variableli
3.1 Introduction..................
3.2 Binomial and Hypergeometric Probabilities
3.3 Discrete Random Variables . .
3.4 Continuous Random Variables
3.5 Expected Values.
3.6 The Variance. . . . . . . . . . .
3.7 Joint Densities . . . . . . . . . .
3.8 Combining Random Variables.
3.9 Further Properties of the Mean and Variance
3.10 Order Statistics .........
3.11 Conditional Densities. . . . . . . . . . . . . .
3.12 Moment-Generating Functions ........
3.13 Taking a Second Look at Statistics (Intotpreting Means)
Appendix 3.A.1 MINIT AB Applications . . . . . . . . . . . .
4 Special Distributions
4.1 Introduction.......
4.2 The Poisson Distribution
4.3 The Normal Distribution
4.4 The Geometric Distribution
4.5 The Negative Binomial Distribution.
4.6 The Gamma Distribution
4.7 Taking a Second Look at Statistics (Monte Carlo
Simulations)
Appendix 4.A.l MINITAB Applications.
Appendix 4.A.2 A Proof of the Central Limit Theorem
5 Estimation
5.1 Introduction
5.2 Estimating Parameters: The Method of Maximum Likelihood
and the Method of Moments
5.3 Interval Estimation
5.4 Properties of Estimators
5.5 Minimum-Variance Estimators: The Cramer-Rao Lower Bound . 394
5.6 Sufficient Estimators
5.7 Consistency
5.8 Bayesian Estimation
5.9 Taking a Second Look at Statistics (Revisiting the Margin of Error) . 423
Appendix 5.A.1 MINIT AB Applications
6 Hypothesis Testing
6.1 Introduction .
6.2 The Decision Rule.
6.3 Testing Binomial Data-Ho: P = Po
6.4 Type I and Type IT Errors
6.5 A Notion of Optimality: The Generalized Likelihood Ratio
6.6 Taking a Second Look at Statistics (Statistical Significance
versus "Practical" Significance)
7 The Normal Distribution
7.1 Introduction
7.2 Comparing In and J; In
7.3 Deriving the Distribution of %/
7.4 Drawing Inferences About J1
7.5 Drawing Inferences About a 2
7.6 Taking a Second Look at Statistics ("Bad" Estimators) .
Appendix 7.Al MINIT AB Applications
Appendix 7.A2 Some Distribution Results for Y and S2 . . .
Appendix 7.A3 A Proof of Theorem 7.5.2
Appendix 7.A4 A Proof that the One-Sample t Test Is a GLRT
8 Types of Data: A Brief Overview
8.1 Introduction
8.2 Classifying Data
8.3 Taking a Second Look at Statistics (Samples Are Not "Valid")
9 Two-Sample Problems
9.1 Introduction
9.2 Testing Ho: fLx = fLy-The Two-Sample t Test.
9.3 Testing Ho: a = ai-The FTest
9.4 Binomial Data: Testing Ho: px = py
9.5 Confidence Intervals for the Two-Sample Problem
9.6 Taking a Second Look at Statistics (Choosing Samples)
Appendix 9.A.l A Derivation of the Two-Sample t Test
(A Proof of Theorem 9.2.2)
Appendix 9.A.2 MINITAB Applications
10 Goodness-of-Fit Tests
10.1 Introduction
10.2 The Multinomial Distribution
10.3 Goodness-of-Fit Tests: All Parameters Known
10.4 Goodness-of-Fit Tests: Parameters Unknown
10.5 Contingency Tables
10.6 Taking a Second Look at Statistics (Outliers) .
Appendix 1O.A.l MINITAB Applications
U Regression
11.1 Introduction
11.2 The Method of Least Squares
11.3 TheLinearModel
11.4 Covariance and Correlation
11.5 The Bivariate Normal Distribution
11.6 Taking a Second Look at Statistics (How Not to Interpret the Sample
Correlation Coefficient)
Appendix 11.A.l MINITAB Applications
Appendix 11.A.2 A Proof of Theorem
12 The Analysis of Variance
12.1 Introduction
12.2 The F Test
12.3 Multiple Comparisons: Tukey's Method
12.4 Testing Subhypotheses with Contrasts
12.5 Data Transformations
12.6 Taking a Second Look at Statistics (Putting the Subject of Statistics
Together-The Contributions of Ronald A. Fisher)
Appendix 12.A.l MINITAB Applications
Appendix 12.A.2 A Proof of Theorem
Appendix 12.A.3
13 Randomized Block Designs
13.1 Introduction
13.2 The FTest for a Randomized Block Design
13.3 The Paired t Test
13.4 Taking a Second Look at Statistics (Choosing Between a Two-Sample
t Test and a Paired t Test)
Appendix 13.A.l MINITAB Applications .
14 Nonparametric Statistics
14.1 Introduction
14.2 The Sign Test
14.3 Wilcoxon Tests.
14.4 The Kruskal-Wallis Test
14.5 The Friedman Test
14.6 Testing for Randomness
14.7 Taking a Second Look at Statistics (Comparing Parametric and Nonparametric Procedures)
Appendix 14.A.1 MINITAB Applications
Appendix: Statistical Tables
Answers to Selected Odd-Numbered Questions
Bibliography
Index
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