Stochastic Modeling and Mathematical Statistics

Abstract

The Calculus of Probability A Bit of Background Approaches to Modeling Randomness The Axioms of Probability Conditional Probability Bayes' Theorem Independence Counting Chapter Problems Discrete Probability Models Random Variables Mathematical Expectation The Hypergeometric Model A Brief Tutorial on Mathematical Induction (Optional) The Binomial Model The Geometric and Negative Binomial Models The Poisson Model Moment-Generating Functions Chapter Problems Continuous Probability Models Continuous Random Variables Mathematical Expectation for Continuous Random Variables Cumulative Distribution Functions The Gamma Model The Normal Model Other Continuous Models Chapter Problems Multivariate Models Bivariate Distributions More on Mathematical Expectation Independence The Multinomial Distribution (Optional) The Multivariate Normal Distribution Transformation Theory Order Statistics Chapter Problems Limit Theorems and Related Topics Chebyshev's Inequality and Its Applications Convergence of Distribution Functions The Central Limit Theorem The Delta Method Theorem Chapter Problems Statistical Estimation: Fixed Sample Size Theory Basic Principles Further Insights into Unbiasedness Fisher Information, the Cram'er-Rao Inequality, and Best Unbiased Estimators Sufficiency, Completeness, and Related Ideas Optimality within the Class of Linear Unbiased Estimators Beyond Unbiasedness Chapter Problems Statistical Estimation: Asymptotic Theory Basic Principles The Method of Moments Maximum Likelihood Estimation A Featured Example: Maximum Likelihood Estimation of the Risk of Disease Based on Data from a Prospective Study of Disease The Newton-Raphson Algorithm A Featured Example: Maximum Likelihood Estimation from Incomplete Data via the EM Algorithm Chapter Problems Interval Estimation Exact Confidence Intervals Approximate Confidence Intervals Sample Size Calculations Tolerance Intervals (Optional) Chapter Problems The Bayesian Approach to Estimation The Bayesian Paradigm Deriving Bayes Estimators Exploring the Relative Performance of Bayes and Frequentist Estimators A Theoretical Framework for Comparing Bayes vs. Frequentist Estimators Bayesian Interval Estimation Chapter Problems Hypothesis Testing Basic Principles Standard Tests for Means and Proportions Sample Size Requirements for Achieving Pre-specified Power Optimal Tests: The Neyman-Pearson Lemma Likelihood Ratio Tests Testing the Goodness of Fit of a Probability Model Fatherly Advice about the Perils of Hypothesis Testing (Optional) Chapter Problems Estimation and Testing for Linear Models Simple Linear Regression Some Distribution Theory for Simple Linear Regression Theoretical Properties of Estimators and Tests under the SLR Model One-Way Analysis of Variance The Likelihood Ratio Test in One-Way ANOVA Chapter Problems Nonparametric Statistical Methods Nonparametric Estimation The Nonparametric Bootstrap The Sign Test The Runs Test The Rank Sum Test Chapter Problems Tables Bibliography Index