Mathematical Statistics: An Introduction to Likelihood Based Inference

Presents a unified approach to parametric estimation, confidence intervals, hypothesis testing, and statistical modeling, which are uniquely based on the likelihood function

This book addresses mathematical statistics for upper–undergraduates and first year graduate students, tying chapters on estimation, confidence intervals, hypothesis testing, and statistical models together to present a unifying focus on the likelihood function. It also emphasizes the important ideas in statistical modeling, such as sufficiency, exponential family distributions, and large sample properties. Mathematical Statistics: An Introduction to Likelihood Based Inference makes advanced topics accessible and understandable and covers many topics in more depth than typical mathematical statistics textbooks. It includes numerous examples, case studies, a large number of exercises ranging from drill and skill to extremely difficult problems, and many of the important theorems of mathematical statistics along with their proofs.

In addition to the connected chapters mentioned above, Mathematical Statistics covers likelihood–based estimation, with emphasis on multidimensional parameter spaces and range dependent support. It also includes a chapter on confidence intervals, which contains examples of exact confidence intervals along with the standard large sample confidence intervals based on the MLE′s and bootstrap confidence intervals. There s also a chapter on parametric statistical models featuring sections on non–iid observations, linear regression, logistic regression, Poisson regression, and linear models.

Prepares students with the tools needed to be successful in their future work in statistics data science Includes practical case studies including real–life data collected from Yellowstone National Park, the Donner party, and the Titanic voyage Emphasizes the important ideas to statistical modeling, such as sufficiency, exponential family distributions, and large sample properties Includes sections on Bayesian estimation and credible intervals Features examples, problems, and solutions 

Mathematical Statistics: An Introduction to Likelihood Based Inference is an ideal textbook for upper–undergraduate and graduate courses in probability, mathematical statistics, and/or statistical inference.

Dedication

Preface

Chapter 1 Probability

1.1   Sample Spaces, Events, and Algebras

Problems

1.2   Probability Axioms and Rules

Problems

1.3   Probability with Equally Likely Outcomes

Problems

1.4   Conditional Probability

Problems

1.5   Independence

Problems

1.6   Counting Methods

Problems

1.7 Case Study The Birthday Problem

Problems

Chapter 2 Random Variables and Random Vectors

2.1 Random Variables

2.1.1 Properties of Random Variables

Problems

2.2 Random Vectors

2.2.1 Properties of Random Vectors

Problems

2.3 Independent Random Variables

Problems

2.4 Transformations of Random Variables

2.4.1 Transformations of Discrete Random Variables

2.4.2 Transformations of Continuous Random Variables

2.4.3 Transformations of Continuous Bivariate Random Vectors

Problems

2.5 Expected Values for Random Variables

2.5.1 Expected Values and Moments of Random Variables

2.5.2 The Variance of a Random Variable

2.5.3 Moment Generating Functions

Problems

2.6 Expected Values for Random Vectors

2.6.1 Properties of Expectation with Random Vectors

2.6.2 Covariance and Correlation

2.6.3 Conditional Expectation and Variance

Problems

2.7 Sums of Random Variables

Problems

2.8 Case Study How Many Times was the Coin Tossed?

2.8.1 The Probability Model

Problems

Chapter 3 Probability Models

3.1 Discrete Probability Models

3.1.1 The Binomial Model

3.1.2 The Hypergeometric Model

3.1.3 The Poisson Model

3.1.4 The Negative Binomial Model

3.1.5 The MultinomialModel

Problems

3.2 Continuous ProbabilityModels

3.2.1 The UniformModel

3.2.2 The Gamma Model

3.2.3 The Normal Model

3.2.4 The LognormalModel

3.2.5 The Beta Model

Problems

3.3 Important Distributional Relationships

3.3.1 Sums of Random Variables

3.3.2 The T and F Distributions

Problems

3.4 Case Study The Central Limit Theorem

3.4.1 Convergence in Distribution

3.4.2 The Central Limit Theorem

Problems

Chapter 4 Parametric Point Estimation

4.1 Statistics

4.1.1 Sampling Distributions

4.1.2 Unbiased Statistics and Estimators

4.1.3 Standard Error and Mean Squared Error

4.1.4 The Delta Method

Problems

4.2 Sufficient Statistics

4.2.1 Exponential Family Distributions

Problems

4.3 Minimum Variance Unbiased Estimators

4.3.1 Cram´erRao Lower Bound

Problems

4.4 Case Study The Order Statistics

Problems

Chapter 5 Likelihood Based Estimation

5.1 Maximum Likelihood Estimation

5.1.1 Properties of MLE s

5.1.2 One Parameter Probability Models

5.1.3 Multi Parameter Probability Models

Problems

5.2 Bayesian Estimation

5.2.1 The Bayesian Setting

5.2.2 Bayesian Estimators

Problems

5.3 Interval Estimation

5.3.1 Exact Confidence Intervals

5.3.2 Large Sample Confidence Intervals

5.3.3 Bayesian Credible Intervals

Problems

5.4 Case Study Modeling Obsidian Rind Thicknesses

5.4.1 Finite Mixture Model

Problems

Chapter 6 Hypothesis Testing

6.1 Components of a Hypothesis Test

Problems

6.2 Most Powerful Tests

Problems

6.3 Uniformly Most Powerful Tests

6.3.1 UniformlyMost Powerful Unbiased Tests

Problems

6.4 Generalized Likelihood Ratio Tests

Problems

6.5 Large Sample Tests

6.5.1 Large Sample Tests based on the MLE

6.5.2 Score Tests

Problems

6.6 Case Study Modeling Survival of the Titanic Passengers

6.6.1 Exploring the Data

6.6.2 Modeling the Probability of Survival

6.6.3 Analysis of the Fitted Survival Model

Problems

Chapter 7 Generalized Linear Models

7.1 Generalized Linear Models

Problems

7.2 Fitting a Generalized Linear Model

7.2.1 Estimating ∼

7.2.2 Model Deviance

Problems

7.3 Hypothesis Testing in a Generalized Linear Model

7.3.1 Asymptotic Properties

7.3.2 Wald Tests and Confidence Intervals

7.3.3 Likelihood Ratio Tests

Problems

7.4 Generalized Linear Models for a Normal Response Variable

7.4.1 Estimation

7.4.2 Properties of the MLE s

7.4.3 Deviance

7.4.4 Hypothesis Testing

Problems

7.5 Generalized Linear Models for a Binomial Response Variable

7.5.1 Estimation

7.5.2 Properties of the MLE s

7.5.3 Deviance

7.5.4 Hypothesis Testing

Problems

7.6 Case Study IDNAP

Experiment with Poisson Count Data

7.6.1 The Model

7.6.2 Statistical Methods

7.6.3 Results of the First Experiment

Problems

References

A Probability Models

B Data Sets

Problem Solutions

Index

Richard J. Rossi, PhD, is Director of the Statistics Program and Co–Director of the Data Science Program at Montana Tech of The University of Montana, in Butte, MT. He acted as President of the Montana Chapter of the American Statistical Association in 2001 and as Associate Editor for Biometrics from 1997–2000. He is a member of the American Mathematical Society, the Institute of Mathematical Statistics, and the American Statistical Association.