1st Edition

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

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*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**

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

### 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 Statisticsis 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