1st Edition

# Stochastic Modeling and Mathematical Statistics A Text for Statisticians and Quantitative Scientists

By Francisco J. Samaniego Copyright 2014
622 Pages 68 B/W Illustrations
by Chapman & Hall

622 Pages
by Chapman & Hall

Also available as eBook on:

Provides a Solid Foundation for Statistical Modeling and Inference and Demonstrates Its Breadth of Applicability

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.

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
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
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

### Biography

F. J. Samaniego has served on the faculty of the University of California, Davis, for four decades, teaching upper division courses on probability and mathematical statistics numerous times. In 2002, he received the UCD Academic Senate Distinguished Teaching Award. He was the 2004 recipient of the Davis Prize for Undergraduate Teaching and Scholarly Achievement.

"Stochastic Modeling and Mathematical Statistics is a new and welcome addition to the corpus of undergraduate statistical textbooks in the market. The singular thing that struck me when I initially perused the book was its lucid and endearing conversational tone, which pervades the entire text. It radiated warmth. … In my course at the University of Michigan, I rely primarily on my own lecture notes and have used Rice as supplementary material. Having gone through this text, I am strongly inclined to add this to the supplementary list as well. I have little doubt that this book will be very successful as a course textbook in the years to come."
International Statistical Review, 82, 2014