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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.

Statistical Inference

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.