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

    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
    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
    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


    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