Proschan / Shaw Essentials of Probability Theory for Statisticians

Erscheinungsjahr 2016
ISBN: 978-1-4987-0422-9
Verlag: CRC Press
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 344 Seiten

Reihe: Chapman & Hall/CRC Texts in Statistical Science

Essentials of Probability Theory for Statisticians provides graduate students with a rigorous treatment of probability theory, with an emphasis on results central to theoretical statistics. It presents classical probability theory motivated with illustrative examples in biostatistics, such as outlier tests, monitoring clinical trials, and using adaptive methods to make design changes based on accumulating data. The authors explain different methods of proofs and show how they are useful for establishing classic probability results.

After building a foundation in probability, the text intersperses examples that make seemingly esoteric mathematical constructs more intuitive. These examples elucidate essential elements in definitions and conditions in theorems. In addition, counterexamples further clarify nuances in meaning and expose common fallacies in logic.

This text encourages students in statistics and biostatistics to think carefully about probability. It gives them the rigorous foundation necessary to provide valid proofs and avoid paradoxes and nonsensical conclusions.

Autoren/Hrsg.

Proschan, Michael A.

Shaw, Pamela A.

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Bayesianische Inferenz

Inhaltsverzeichnis

Introduction

Why More Rigor Is Needed

Size Matters

Cardinality

Summary

The Elements of Probability Theory

Introduction

Sigma-Fields

The Event That An Occurs Infinitely Often

Measures/Probability Measures

Why Restriction of Sets Is Needed

When We Cannot Sample Uniformly

The Meaninglessness of Post-Facto Probability Calculations

Summary

Random Variables and Vectors

Random Variables

Random Vectors

The Distribution Function of a Random Variable

The Distribution Function of a Random Vector

Introduction to Independence

Take (O, F, P) = ((0, 1), B(0,1), µL), Please!

Summary

Integration and Expectation

Heuristics of Two Different Types of Integrals

Lebesgue–Stieltjes Integration

Properties of Integration

Important Inequalities

Iterated Integrals and More on Independence

Densities

Keep It Simple

Summary

Modes of Convergence

Convergence of Random Variables

Connections between Modes of Convergence

Convergence of Random Vectors

Summary

Laws of Large Numbers

Basic Laws and Applications

Proofs and Extensions

Random Walks

Summary

Central Limit Theorems

CLT for iid Random Variables and Applications

CLT for Non iid Random Variables

Harmonic Regression

Characteristic Functions

Proof of Standard CLT

Multivariate Ch.f.s and CLT

Summary

More on Convergence in Distribution

Uniform Convergence of Distribution Functions

The Delta Method

Convergence of Moments: Uniform Integrability

Normalizing Sequences

Review of Equivalent Conditions for Weak Convergence

Summary

Conditional Probability and Expectation

When There Is a Density or Mass Function
More General Definition of Conditional Expectation

Regular Conditional Distribution Functions

Conditional Expectation as a Projection

Conditioning and Independence

Sufficiency

Expect the Unexpected from Conditional Expectation

Conditional Distribution Functions as Derivatives

Appendix: Radon–Nikodym Theorem

Summary

Applications

F(X) ~ U[0, 1] and Asymptotics

Asymptotic Power and Local Alternatives

Insufficient Rate of Convergence in Distribution

Failure to Condition on All Information

Failure to Account for the Design

Validity of Permutation Tests: I

Validity of Permutation Tests: II

Validity of Permutation Tests III

A Brief Introduction to Path Diagrams

Estimating the Effect Size

Asymptotics of an Outlier Test

An Estimator Associated with the Logrank Statistic

Appendix A: Whirlwind Tour of Prerequisites

Appendix B: Common Probability Distributions

Appendix C: References

Appendix D: Mathematical Symbols and Abbreviations

Index

Über Autor(innen)

Michael A. Proschan is a mathematical statistician in the Biostatistics Research Branch at the U.S. National Institute of Allergy and Infectious Diseases (NIAID). A fellow of the American Statistical Association, Dr. Proschan has published more than 100 articles in numerous peer-reviewed journals. His research interests include monitoring clinical trials, adaptive methods, permutation tests, and probability. He earned a PhD in statistics from Florida State University.

Pamela A. Shaw is an assistant professor of biostatistics in the Department of Biostatistics and Epidemiology at the University of Pennsylvania Perelman School of Medicine. Dr. Shaw has published several articles in numerous peer-reviewed journals. Her research interests include methodology to address covariate and outcome measurement error, the evaluation of diagnostic tests, and the design of medical studies. She earned a PhD in biostatistics from the University of Washington.