Stochastic Modeling and Mathematical Statistics: A Text for Statisticians and Quantitative Scientists addresses core issues in post-calculus probability and statistics in a way that is useful for statistics and mathematics majors as well as students in the quantitative sciences. The book's conversational tone, which provides the mathematical justification behind widely used statistical methods in a reader-friendly manner, and the book's many examples, tutorials, exercises and problems for solution, together constitute an effective resource that students can read and learn from and instructors can count on as a worthy complement to their lectures. Provides a Solid Foundation for Statistical Modeling and Inference and Demonstrates its Breadth of Applicability Carefully introduces the probability models and tools that are essential for studying statistical theory and applications Treats statistical estimation from a host of different perspectives, with a full chapter on the Bayesian approach and a section on nonparametric estimation Details both the theory and practice of hypothesis testing, accompanied by careful advice about potential misuses Shows how estimation and testing theory may be applied to data obeying regression or ANOVA models Includes sections on Newton-Raphson iterations, on the EM algorithm, on odds ratio estimation in cohort studies and on the nonparametric bootstrap Using classroom-tested approaches that engage students in active learning, the text offers instructors the flexibility to control the mathematical level of their course. It contains the mathematical detail that is expected in a course for "majors" but is written in a way that emphasizes the intuitive content in statistical theory and the way theoretical results are used in practice. More than 1000 exercises and problems at varying levels of difficulty and with a broad range of topical focus give instructors many options in assigning homework and provide students with many problems on which to practice and from which to learn.
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
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