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This book builds theoretical statistics from the first principles of probability theory. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and are natural extensions and consequences of previous concepts. Intended for first-year graduate students, this book can be used for students majoring in statistics who have a solid mathematics background. It can also be used in a way that stresses the more practical uses of statistical theory, being more concerned with understanding basic statistical concepts and deriving reasonable statistical procedures for a variety of situations, and less concerned with formal optimality investigations.

1. Probability Theory.

Set Theory. Probability Theory. Conditional Probability and Independence. Random Variables. Distribution Functions. Density and Mass Functions. Exercises. Miscellanea.

2. Transformations and Expectations.

Distribution of Functions of a Random Variable. Expected Values. Moments and Moment Generating Functions. Differentiating Under an Integral Sign. Exercises. Miscellanea.

3. Common Families of Distributions.

Introductions. Discrete Distributions. Continuous Distributions. Exponential Families. Locations and Scale Families. Inequalities and Identities. Exercises. Miscellanea.

4. Multiple Random Variables.

Joint and Marginal Distributions. Conditional Distributions and Independence. Bivariate Transformations. Hierarchical Models and Mixture Distributions. Covariance and Correlation. Multivariate Distributions. Inequalities. Exercises. Miscellanea.

5. Properties of a Random Sample.

Basic Concepts of Random Samples. Sums of Random Variables from a Random Sample. Sampling for the Normal Distribution. Order Statistics. Convergence Concepts. Generating a Random Sample. Exercises. Miscellanea.

6. Principles of Data Reduction.

Introduction. The Sufficiency Principle. The Likelihood Principle. The Equivariance Principle. Exercises. Miscellanea.

7. Point Estimation.

Introduction. Methods of Finding Estimators. Methods of Evaluating Estimators. Exercises. Miscellanea.

8. Hypothesis Testing.

Introduction. Methods of Finding Tests. Methods of Evaluating Test. Exercises. Miscellanea.

9. Interval Estimation.

Introduction. Methods of Finding Interval Estimators. Methods of Evaluating Interval Estimators. Exercises. Miscellanea.

10. Asymptotic Evaluations.

Point Estimation. Robustness. Hypothesis Testing. Interval Estimation. Exercises. Miscellanea.

11. Analysis of Variance and Regression.

Introduction. One-way Analysis of Variance. Simple Linear Regression. Exercises. Miscellanea.

12. Regression Models.

Introduction. Regression with Errors in Variables. Logistic Regression. Robust Regression. Exercises. Miscellanea. Appendix. Computer Algebra. References.

- Offers new coverage of random number generation, simulation methods, bootstrapping, EM algorithm, p-values, and robustness.
- Gathers all large sample results into Chapter 10.
- Includes a new section on "Generating a Random Sample" in Chapter 5.
- Includes new sections on "Logistic Regression" and "Robust Regression" in Chapter 12.
- Contains updated and expanded Exercises in all chapters, and updated and expanded Miscellanea including discussions of variations on likelihood and Bayesian analysis, bootstrap, "second-order" asymptotics, and Monte Carlo Markov chain.
- Contains an Appendix detailing the use of Mathematica in problem solving.
- Restructures material for clarity purposes.

- Begins with the basics of probability theory and introduces many fundamentals that are later necessary (Chapters 1-4).
- Treats likelihood and sufficiency principles in detail. These principles, and the thinking behind them, are fundamental to total statistical understanding. The equivariance principle is also introduced.
- Divides the methods of finding appropriate statistical methods and the methods of evaluating these techniques in the core statistical inference chapters (Chapters 7-9). Integrates decision theoretic evaluations into core chapters. Many of the techniques are used in consulting and are helpful in analyzing and inferring from actual problems.
- Discusses use of simulation in mathematical statistics.
- Includes a thorough introduction to large sample statistical methods.
- Covers the elementary linear models through simple linear regression and oneway analysis of variance.
- Covers more advanced theory of regression topics including "errors in variables" regression, logistic regression, and robust regression.

**George Casella**

University of Florida

**Roger L. Berger**

Arizona State University

"Statistical Inference is a delightfully modern text on statistical theory and deserves serious consideration from every teacher of a graduate- or advanced undergraduate-level first course in statistical theory. . . Chapters 1-5 provide plenty of interesting examples illustrating either the basic concepts of probability or the basic techniques of finding distribution. . . The book has unique features [throughout Chapters 6-12] for example, I have never seen in any comparable text such extensive discussion of ancillary statistics [Ch. 6], including Basu's theorem, dealing with the independence of complete sufficient statistics and ancillary statistics. Basu's theorem is such a useful tool that it should be available to every graduate student of statistics. . . The derivation of the analysis of variance (ANOVA)F test in Chapter 11 via the union-intersection principle is very nice. . . Chapter 12 contains, in addition to the standard regression model, errors-in-variables models. This topic will be of considerable importance in the years ahead, and the authors should be thanked for giving the reader an introduction to it. . . Another nice feature is the Miscellanea Section at the end of nearly every chapter. This gives the serious student an opportunity to go beyond the basic material of the text and look at some of the more advanced work on the topics, thereby developing a much better feel for the subject."

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