Statistical Inference

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. This book can be used for readers 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. TRANSFORMATION 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 EXTIMATION
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

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Statistical Inference

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