Introduction to Mathematical Statistics

For courses in mathematical statistics.

Comprehensive coverage of mathematical statistics — with a proven approach

Introduction to Mathematical Statistics by Hogg, McKean, and Craig enhances student comprehension and retention with numerous, illustrative examples and exercises. Classical statistical inference procedures in estimation and testing are explored extensively, and the text’s flexible organization makes it ideal for a range of mathematical statistics courses.

Substantial changes to the 8th Edition — many based on user feedback — help students appreciate the connection between statistical theory and statistical practice, while other changes enhance the development and discussion of the statistical theory presented. 

0134686993 / 9780134686998  Introduction to Mathematical Statistics, 8/e

About our authors

Robert V. Hogg (deceased), Professor Emeritus of Statistics at the University of Iowa since 2001, received his B.A. in mathematics at the University of Illinois and his M.S. and Ph.D. degrees in mathematics, specializing in actuarial sciences and statistics, from the University of Iowa. Known for his gift of humor and his passion for teaching, Hogg had far-reaching influence in the field of statistics. Throughout his career, Hogg played a major role in defining statistics as a unique academic field, and he almost literally "wrote the book" on the subject. He wrote more than 70 research articles and co-authored four books, including Introduction of Mathematical Statistics, 6th Edition with J. W. McKean and A.T. Craig; Applied Statistics for Engineers and Physical Scientists, 3rd Edition with J. Ledolter; and A Brief Course in Mathematical Statistics, 1st Edition with E.A. Tanis. His texts have become classroom standards used by hundreds of thousands of students.

Among the many awards he received for distinction in teaching, Hogg was honored at the national level (the Mathematical Association of America Award for Distinguished Teaching), the state level (the Governor's Science Medal for Teaching), and the university level (Collegiate Teaching Award). His important contributions to statistical research have been acknowledged by his election to fellowship standing in the ASA and the Institute of Mathematical Statistics.

Elliot Tanis, Professor Emeritus of Mathematics at Hope College, received his M.S. and Ph.D. degrees from the University of Iowa. Tanis is the co-author of A Brief Course in Mathematical Statistics, 1st Edition with R. Hogg and Probability and Statistics: Explorations with MAPLE, 2nd Edition with Z. Karian. He has authored over 30 publications on statistics and is a past chairman and governor of the Michigan MAA, which presented him with both its Distinguished Teaching and Distinguished Service Awards. He taught at Hope for 35 years and in 1989 received the HOPE Award (Hope's Outstanding Professor Educator) for his excellence in teaching. In addition to his academic interests, Dr. Tanis is also an avid tennis player and devoted Hope sports fan.

Dale Zimmerman is the Robert V. Hogg Professor in the Department of Statistics and Actuarial Science at the University of Iowa.

(Note: Sections marked with an asterisk * are optional.)

1. Probability and Distributions 

1.1 Introduction 

1.2 Sets 

1.3 The Probability Set Function 

1.4 Conditional Probability and Independence 

1.5 Random Variables 

1.6 Discrete Random Variables 

1.7 Continuous Random Variables 

1.8 Expectation of a Random Variable

1.9 Some Special Expectations 

1.10 Important Inequalities

2. Multivariate Distributions

2.1 Distributions of Two Random Variables 

2.2 Transformations: Bivariate Random Variables 

2.3 Conditional Distributions and Expectations 

2.4 Independent Random Variables

2.5 The Correlation Coefficient 

2.6 Extension to Several Random Variables 

2.7 Transformations for Several Random Variables 

2.8 Linear Combinations of Random Variables 

3. Some Special Distributions

3.1 The Binomial and Related Distributions 

3.2 The Poisson Distribution 

3.3 The Γ, χ2, and β Distributions 

3.4 The Normal Distribution 

3.5 The Multivariate Normal Distribution

3.6 t- and F-Distributions

3.7 Mixture Distributions*

4. Some Elementary Statistical Inferences

4.1 Sampling and Statistics

4.2 Confidence Intervals 

4.3 ∗Confidence Intervals for Parameters of Discrete Distributions 

4.4 Order Statistics 

4.5 Introduction to Hypothesis Testing 

4.6 Additional Comments About Statistical Tests 

4.7 Chi-Square Tests 

4.8 The Method of Monte Carlo 

4.9 Bootstrap Procedures 

4.10 Tolerance Limits for Distributions* 

5. Consistency and Limiting Distributions

5.1 Convergence in Probability 

5.2 Convergence in Distribution 

5.3 Central Limit Theorem 

5.4 Extensions to Multivariate Distributions* 

6. Maximum Likelihood Methods

6.1 Maximum Likelihood Estimation 

6.2 Rao—Cramér Lower Bound and Efficiency 

6.3 Maximum Likelihood Tests 

6.4 Multiparameter Case: Estimation

6.5 Multiparameter Case: Testing 

6.6 The EM Algorithm 

7. Sufficiency

7.1 Measures of Quality of Estimators 

7.2 A Sufficient Statistic for a Parameter

7.3 Properties of a Sufficient Statistic

7.4 Completeness and Uniqueness

7.5 The Exponential Class of Distributions 

7.6 Functions of a Parameter 

7.7 The Case of Several Parameters 

7.8 Minimal Sufficiency and Ancillary Statistics 

7.9 Sufficiency, Completeness, and Independence

8. Optimal Tests of Hypotheses

8.1 Most Powerful Tests 

8.2 Uniformly Most Powerful Tests 

8.3 Likelihood Ratio Tests

8.3.2 Likelihood Ratio Tests for Testing Variances of Normal Distributions

8.4 The Sequential Probability Ratio Test* 

8.5 Minimax and Classification Procedures*

9. Inferences About Normal Linear Models

9.1 Introduction 

9.2 One-Way ANOVA 

9.3 Noncentral χ2 and F-Distributions 

9.4 Multiple Comparisons 

9.5 Two-Way ANOVA 

9.6 A Regression Problem

9.7 A Test of Independence

9.8 The Distributions of Certain Quadratic Forms 

9.9 The Independence of Certain Quadratic Forms 

10. Nonparametric and Robust Statistics

10.1 Location Models 

10.2 Sample Median and the Sign Test 

10.3 Signed-Rank Wilcoxon 

10.4 Mann—Whitney—Wilcoxon Procedure 

10.5 General Rank Scores*

10.6 Adaptive Procedures* 

10.7 Simple Linear Model 

10.8 Measures of Association 

10.9 Robust Concepts

11. Bayesian Statistics

11.1 Bayesian Procedures 

11.2 More Bayesian Terminology and Ideas 

11.3 Gibbs Sampler 

11.4 Modern Bayesian Methods 

Appendices:

A. Mathematical Comments

A.1 Regularity Conditions 

A.2 Sequences 

B. R Primer

B.1 Basics 

B.2 Probability Distributions 

B.3 R Functions 

B.4 Loops

B.5 Input and Output 

B.6 Packages

C. Lists of Common Distributions

D. Table of Distributions

E. References

F. Answers to Selected Exercises

Index

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