Models for Probability and Statistical Inference was written over a five-year period and serves as a comprehensive treatment of the fundamentals of probability and statistical inference. With detailed theoretical coverage found throughout the book, readers acquire the fundamentals needed to advance to more specialized topics, such as sampling, linear models, design of experiments, statistical computing, survival analysis, and bootstrapping.
Ideal as a textbook for a two-semester sequence on probability and statistical inference, early chapters provide coverage on probability and include discussions of: discrete models and random variables; discrete distributions including binomial, hypergeometric, geometric, and Poisson; continuous, normal, gamma, and conditional distributions; and limit theory. Since limit theory is usually the most difficult topic for readers to master, the author thoroughly discusses modes of convergence of sequences of random variables, with special attention to convergence in distribution. The second half of the book addresses statistical inference, beginning with a discussion on point estimation and followed by coverage of consistency and confidence intervals. Further areas of exploration include: distributions defined in terms of the multivariate normal, chi-square, t, and F (central and non-central); the one- and two-sample Wilcoxon test, together with methods of estimation based on both; linear models with a linear space-projection approach; and logistic regression.
Each section contains a set of problems ranging in difficulty from simple to more complex, and selected answers as well as proofs to almost all statements are provided. An abundant amount of figures in addition to helpful simulations and graphs produced by the statistical package S-Plus(r) are included to help build the intuition of readers.
1.1 Discrete Probability Models.
1.2 Conditional Probability and Independence.
1.3 Random Variables.
1.5 The Variance.
1.6 Covariance and Correlation.
2. Special Discrete Distributions.
2.1 The Binomial Distribution.
2.2 The Hypergeometric Distribution.
2.3 The Geometric and Negative Binomial Distributions.
2.4 The Poisson Distribution.
3. Continuous Random Variables.
4.1 Continuous RV's and Their Distributions.
4.2 Expected Values and Variances.
4.3 Transformations of Random Variables.
4 Special Continuous Distributions.
4.1 The Normal Distribution.
4.2 The Gamma Distribution.
5. Conditional Distributions.
5.1 The Discrete Case.
5.2 Conditional Expectations for the Discrete Case.
5.3 Conditional Densities and Expectations for Continuous RV's.
6. Limit Laws.
6.1 Moment Generating Functions.
6.2 Convergence in Probability and in Distribution.
6.3 The Central Limit Theorem.
6.4 The Delta-Method.
7.1 Point Estimation.
7.2 The Method of Moments.
7.3 Maximum Likelihood.
7.5 The Ω-Method.
7.6 Confidence Intervals.
7.7 Fisher Information, The Cramer-Rao Bound, and Asymptotic Normality of MLE's.
8. Testing Hypotheses.
8.2 The Neyman-Pearson Lemma.
8.3 The Likelihood Ratio Test.
8.4 The p-Value and the Relationship Between Tests of Hypotheses and Confidence Intervals.
9. The Multivariate Normal, Chi-square, t, and F-Distributions.
9.1 The Multivariate Normal Distribution.
9.2 The Central and Noncentral Chi-Square Distributions.
9.3 Student's t-Distribution.
9.4 The F-Distribution.
10.3 Nonparametric Statistics.
10.1 The Wilcoxon Test and Estimator.
10.2 One Sample Methods.
10.3 The Kolmogorov-Smirnov Tests.
11. Linear Models.
11.1 The Principle of Least Squares.
11.2 Linear Models.
11.3 F-Tests for H0.
11.4 Two-Way Analysis of Variance..
12. Frequency Data.
12.1 Logistic Regression.
12.2 Two-Way Frequency Tables.
12.3 Chi-Square Goodness of Fit Tests.
13. Miscellaneous Topics.
13.1 Survival Analysis.
13.3 Bayesian Statistics.