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El. knyga: Essentials of Probability Theory for Statisticians

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, (National Institute of Allergy and Infectious Diseases (NIAID), National Institutes of Health (NIH), Bethesda, Maryland, USA)
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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.

Recenzijos

" The book Essentials of Probability Theory for Statisticians does not try to compete with probability textbooks like Billingsley (2012) or Chung (2001), but targets a particular audience: graduate students in statistics who need to quickly learn the essentials of probability theory to make rigorous arguments in statistics. The book does not try to give a full introduction to measure theory but instead focuses on the essentials that are needed by statisticians. . . . I think that this textbook fills an important niche: It provides a concise summary of the essentials of probability theory that are needed by statisticians and at the same time relates these concepts to important applications in statistics. Hence, the reader learns to appreciate the interplay between probability and statistics. The book is well written and uses engaging language and plenty of examples and illustrations. Overall, I enjoyed teaching from this book and plan to use it again for future graduate-level teaching in statistics." Journal of the American Statistical Association

"This book has tremendous potential for usage in statistics and biostatistics departments where the Ph.D. students would not necessarily have taken a measure theory course but would need a rigorous treatment of probability for their dissertation research and publications in statistical and biostatistics journals The authors are commended for providing this valuable book for students in statistics and biostatistics. The illustrative biostatistics examples (throughout chapter 10 but especially in chapter 11) provide motivating rewards for students." Robert Taylor, Clemson University

" a very good textbook choice for our courses on advanced probability theory (I, II) at the graduate level." Jie Yang, University of Illinois at Chicago

"Many successful graduate students in statistics lack the mathematical prerequisites necessary for Billingsleys book and find su

Preface xiii
Index of Statistical Applications and Notable Examples xv
1 Introduction 1(8)
1.1 Why More Rigor is Needed
1(8)
2 Size Matters 9(8)
2.1 Cardinality
9(7)
2.2 Summary
16(1)
3 The Elements of Probability Theory 17(22)
3.1 Introduction
17(1)
3.2 Sigma-Fields
18(8)
3.2.1 General Case
18(2)
3.2.2 Sigma-fields When Ω = R
20(2)
3.2.3 Sigma-fields When Ω = Rk
22(4)
3.3 The Event That An Occurs Infinitely Often
26(1)
3.4 Measures/Probability Measures
27(6)
3.5 Why Restriction of Sets is Needed
33(2)
3.6 When We Cannot Sample Uniformly
35(1)
3.7 The Meaninglessness of Post-Facto Probability Calculations
36(1)
3.8 Summary
36(3)
4 Random Variables and Vectors 39(36)
4.1 Random Variables
39(7)
4.1.1 Sigma-Fields Generated by Random Variables
39(4)
4.1.2 Probability Measures Induced by Random Variables
43(3)
4.2 Random Vectors
46(2)
4.2.1 Sigma-Fields Generated by Random Vectors
46(2)
4.2.2 Probability Measures Induced by Random Vectors
48(1)
4.3 The Distribution Function of a Random Variable
48(4)
4.4 The Distribution Function of a Random Vector
52(4)
4.5 Introduction to Independence
56(10)
4.5.1 Definitions and Results
56(7)
4.5.2 Product Measure
63(3)
4.6 Take (Ω, F, P) = ((0, 1), B(0,1), μL), Please!
66(6)
4.7 Summary
72(3)
5 Integration and Expectation 75(26)
5.1 Heuristics of Two Different Types of Integrals
75(3)
5.2 Lebesgue-Stieltjes Integration
78(3)
5.2.1 Nonnegative Integrands
78(2)
5.2.2 General Integrands
80(1)
5.3 Properties of Integration
81(6)
5.4 Important Inequalities
87(3)
5.5 Iterated Integrals and More on Independence
90(4)
5.6 Densities
94(3)
5.7 Keep It Simple
97(2)
5.8 Summary
99(2)
6 Modes of Convergence 101(28)
6.1 Convergence of Random Variables
102(12)
6.1.1 Almost Sure Convergence
102(3)
6.1.2 Convergence in Probability
105(4)
6.1.3 Convergence in Lp
109(1)
6.1.4 Convergence in Distribution
110(4)
6.2 Connections between Modes of Convergence
114(12)
6.2.1 Convergence in Lp Implies Convergence in Probability, but Not Vice Versa
115(1)
6.2.2 Convergence in Probability Implies Convergence in Distribution, but Not Vice Versa
116(1)
6.2.3 Almost Sure Convergence Implies Convergence in Probability, but Not Vice Versa
117(3)
6.2.4 Subsequence Arguments
120(4)
6.2.5 Melding Modes: Slutsky's Theorem
124(2)
6.3 Convergence of Random Vectors
126(2)
6.4 Summary
128(1)
7 Laws of Large Numbers 129(20)
7.1 Basic Laws and Applications
129(9)
7.2 Proofs and Extensions
138(4)
7.3 Random Walks
142(6)
7.4 Summary
148(1)
8 Central Limit Theorems 149(34)
8.1 CLT for iid Random Variables and Applications
149(6)
8.2 CLT for Non iid Random Variables
155(8)
8.3 Harmonic Regression
163(3)
8.4 Characteristic Functions
166(7)
8.5 Proof of Standard CLT
173(3)
8.5.1 Proof of CLT for Symmetric Random Variables
174(1)
8.5.2 Proof of CLT for Arbitrary Random Variables
174(2)
8.6 Multivariate Ch.f.s and CLT
176(6)
8.7 Summary
182(1)
9 More on Convergence in Distribution 183(18)
9.1 Uniform Convergence of Distribution Functions
183(5)
9.2 The Delta Method
188(5)
9.3 Convergence of Moments: Uniform Integrability
193(3)
9.4 Normalizing Sequences
196(1)
9.5 Review of Equivalent Conditions for Weak Convergence
197(2)
9.6 Summary
199(2)
10 Conditional Probability and Expectation 201(40)
10.1 When There is a Density or Mass Function
202(3)
10.2 More General Definition of Conditional Expectation
205(6)
10.3 Regular Conditional Distribution Functions
211(6)
10.4 Conditional Expectation As a Projection
217(3)
10.5 Conditioning and Independence
220(3)
10.6 Sufficiency
223(6)
10.6.1 Sufficient and Ancillary Statistics
223(1)
10.6.2 Completeness and Minimum Variance Unbiased Estimation
224(1)
10.6.3 Basu's Theorem and Applications
225(1)
10.6.4 Conditioning on Ancillary Statistics
226(3)
10.7 Expect the Unexpected from Conditional Expectation
229(8)
10.7.1 Conditioning on Sets of Probability 0
230(1)
10.7.2 Substitution in Conditioning Expressions
231(4)
10.7.3 Weak Convergence of Conditional Distributions
235(2)
10.8 Conditional Distribution Functions As Derivatives
237(2)
10.9 Appendix: Radon-Nikodym Theorem
239(1)
10.10 Summary
239(2)
11 Applications 241(38)
11.1 F(X)~U[ 0,1] and Asymptotics
241(2)
11.2 Asymptotic Power and Local Alternatives
243(3)
11.2.1 T-Test
244(1)
11.2.2 Test of Proportions
245(1)
11.2.3 Summary
246(1)
11.3 Insufficient Rate of Convergence in Distribution
246(2)
11.4 Failure to Condition on All Information
248(1)
11.5 Failure to Account for the Design
249(4)
11.5.1 Introduction and Simple Analysis
249(1)
11.5.2 Problem with the Proposed Method
250(1)
11.5.3 Connection to Fisher's Exact Test
250(1)
11.5.4 A Flawed Argument that P(Zn1 < Z1|Zn2 = z2)-> Φ(zi)
251(1)
11.5.5 Fixing the Flaw: Polya's Theorem to the Rescue
251(1)
11.5.6 Conclusion: Asymptotics of the Hypergeometric Distribution
252(1)
11.6 Validity of Permutation Tests: I
253(2)
11.7 Validity of Permutation Tests: II
255(3)
11.7.1 A Vaccine Trial Raising Validity Questions
255(1)
11.7.2 Assumptions Ensuring Validity
255(2)
11.7.3 Is a Z-Test of Proportions Asymptotically Valid?
257(1)
11.8 Validity of Permutation Tests III
258(3)
11.8.1 Is Adaptive Selection of Covariates Valid?
258(1)
11.8.2 Sham Covariates Reduce Variance: What's the Rub?
259(1)
11.8.3 A Disturbing Twist
260(1)
11.9 A Brief Introduction to Path Diagrams
261(6)
11.10 Estimating the Effect Size
267(3)
11.10.1 Background and Introduction
267(1)
11.10.2 An Erroneous Application of Slutsky's Theorem
268(1)
11.10.3 A Correct Application of Slutsky's Theorem
268(1)
11.10.4 Transforming to Stabilize The Variance: The Delta Method
269(1)
11.11 Asymptotics of An Outlier Test
270(4)
11.11.1 Background and Test
270(1)
11.11.2 Inequality and Asymptotics for 1 — Φ(x)
271(1)
11.11.3 Relevance to Asymptotics for the Outlier Test
272(2)
11.12 An Estimator Associated with the Logrank Statistic
274(5)
11.12.1 Background and Goal
274(1)
11.12.2 Distributions of (Xi|nTi,nCi) and Xi — E(Xi|nTi,nCi)
275(1)
11.12.3 The Relative Error of τn,
276(3)
A Whirlwind Tour of Prerequisites 279(22)
A.1 A Key Inequality
279(1)
A.2 The Joy of Sets
279(3)
A.2.1 Basic Definitions and Results
279(2)
A.2.2 Sets of Real Numbers: Inf and Sup
281(1)
A.3 A Touch of Topology of Rk
282(6)
A.3.1 Open and Closed Sets
282(4)
A.3.2 Compact Sets
286(2)
A.4 Sequences in Rk
288(3)
A.5 Series
291(3)
A.6 Functions
294(7)
A.6.1 Mappings
294(1)
A.6.2 Limits and Continuity of Functions
295(3)
A.6.3 The Derivative of a Function of k Variables
298(3)
B Common Probability Distributions 301(8)
B.1 Discrete Distributions
301(2)
B.2 Continuous Distributions
303(2)
B.3 Relationships between Distributions
305(4)
C References 309(4)
D Mathematical Symbols and Abbreviations 313(4)
Index 317
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.
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