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Statistical Inference 2th.pdf
Contents
1 Probability Theory
1.1 Set Theory
1.2 Basics of Probability Theory
1.2.1 Axiomatic Foundations
1.2.2 The Calculus of Probabilities
1.2.3 Counting
1.2.4 Enumerating Outcomes
1.3 Conditional Probability and Independence
1.4 Random Variables
1.5 Distribution Functions
1.6 Density and Mass Functions
1.7 Exercises
1.8 Miscellanea
2 Transformations and Expectations
2.1 Distributions of Functions of a Random Variable
2.2 Expected Values
2.3 Moments and Moment Generating Functions
2.4 Differentiating Under an Integral Sign
2.5 Exercises
2.6 Miscellanea
3 Common Families of Distributions
3.1 Introduction
3.2 Discrete Distributions
3.3 Continuous Distributions
3.4 Exponential Families
3.5 Location and Scale Families
3.6 Inequalities and Identities
3.6.1 Probability Inequalities
3.6.2 Identities
3.7 Exercises
3.8 Miscellanea
4 Multiple Random Variables
4.1 Joint and Marginal Distributions
4.2 Conditional Distributions and Independence
4.3 Bivariate Transformations
4.4 Hierarchical Models and Mixture Distributions
4.5 Covariance and Correlation
4.6 Multivariate Distributions
4.7 Inequalities
4.7.1 Numerical Inequalities
4.7.2 Functional Inequalities
4.8 Exercises
4.9 Miscellanea
5 Properties of a Random Sample
5.1 Basic Concepts of Random Samples
5.2 Sums of Random Variables from a Random Sample
5.3 Sampling from the Normal Distribution
5.3.1 Properties of the Sample Mean and Variance
5.3.2 The Derived Distributions: Student's t and Snedecor's F
5.4 Order Statistics
5.5 Convergence Concepts
5.5.1 Convergence in Probability
5.5.2 Almost Sure Convergence
5.5.3 Convergence in Distribution
5.5.4 The Delta Method
5.6 Generating a Random Sample
5.6.1 Direct Methods
5.6.2 Indirect Methods
5.6.3 The Accept/Reject Algorithm
5.7 Exercises
5.8 Miscellanea
6 Principles of Data Reduction
6.1 Introduction
6.2 The Sufficiency Principle
6.2.1 Sufficient Statistics
6.2.2 Minimal Sufficient Statistics
6.2.3 Ancillary Statistics
6.2.4 Sufficient, Ancillary, and Complete Statistics
6.3 The Likelihood Principle
6.3.1 The Likelihood Function
6.3.2 The Formal Likelihood Principle
6.4 The Equivariance Principle
6.5 Exercises
6.6 Miscellanea
7 Point Estimation
7.1 Introduction
7.2 Methods of Finding Estimators
7.2.1 Method of Moments
7.2.2 Maximum Likelihood Estimators
7.2.3 Bayes Estimators
7.2.4 The EM Algorithm
7.3 Methods of Evaluating Estimators
7.3.1 Mean Squared Error
7.3.2 Best Unbiased Estimators
7.3.3 Sufficiency and Unbiasedness
7.3.4 Loss Function Optimality
7.4 Exercises
7.5 Miscellanea
8 Hypothesis Testing
8.1 Introduction
8.2 Methods of Finding Tests
8.2.1 Likelihood Ratio Tests
8.2.2 Bayesian Tests
8.2.3 Union-Intersection and Intersection-Union Tests
8.3 Methods of Evaluating Tests
8.3.1 Error Probabilities and the Power Function
8.3.2 Most Powerful Tests
8.3.3 Sizes of Union-Intersection and Intersection-Union Tests
8.3.4 p-Values
8.3.5 Loss Function Optimality
8.4 Exercises
8.5 Miscellanea
9 Interval Estimation
9.1 Introduction
9.2 Methods of Finding Interval Estimators
9.2.1 Inverting a Test Statistic
9.2.2 Pivotal Quantities
9.2.3 Pivoting the CDF
9.2.4 Bayesian Intervals
9.3 Methods of Evaluating Interval Estimators
9.3.1 Size and Coverage Probability
9.3.2 Test-Related Optimality
9.3.3 Bayesian Optimality
9.3.4 Loss Function Optimality
9.4 Exercises
9.5 Miscellanea
10 Asymptotic Evaluations
10.1 Point Estimation
10.1.1 Consistency
10.1.2 Efficiency
10.1.3 Calculations and Comparisons
10.1.4 Bootstrap Standard Errors
10.2 Robustness
10.2.1 The Mean and the Median
10.2.2 M-Estimators
10.3 Hypothesis Testing
10.3.1 Asymptotic Distribution of LRTs
10.3.2 Other Large-Sample Tests
10.4 Interval Estimation
10.4.1 Approximate Maximum Likelihood Intervals
10.4.2 Other Large-Sample Intervals
10.5 Exercises
10.6 Miscellanea
11 Analysis of Variance and Regression
11.1 Introduction
11.2 Oneway Analysis of Variance
11.2.1 Model and Distribution Assumptions
11.2.2 The Classic ANOVA Hypothesis
11.2.3 Inferences Regarding Linear Combinations of Means
11.2.4 The ANOVA F Test
11.2.5 Simultaneous Estimation of Contrasts
11.2.6 Partitioning Sums of Squares
11.3 Simple Linear Regression
11.3.1 Least Squares: A Mathematical Solution
11.3.2 Best Linear Unbiased Estimators: A Statistical Solution
11.3.3 Models and Distribution Assumptions
11.3.4 Estimation and Testing with Normal Errors
11.3.5 Estimation and Prediction at a Specified x = xo
11.3.6 Simultaneous Estimation and Confidence Bands
11.4 Exercises
11.5 Miscellanea
12 Regression Models
12.1 Introduction
12.2 Regression with Errors in Variables
12.2.1 Functional and Structural Relationships
12.2.2 A Least Squares Solution
12.2.3 Maximum Likelihood Estimation
12.2.4 Confidence Sets
12.3 Logistic Regression
12.3.1 The Model
12.3.2 Estimation
12.4 Robust Regression
12.5 Exercises
12.6 Miscellanea
Appendix: Computer Algebra
Table of Common Distributions
References
Author Index
SUbject Index
Stalisticallnference
Second fdition
George Casella
Roger l. Berger
DUXBURY ADVANCED SER IES
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Statistical Inference
Second Edition
George Casella
University of Florida
Roger L. Berger
North Carolina State University
DUXBURY •
THOMSON LEARNING
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Library of Congress Cataloging-in-P-:;;6tfcation Data
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Statistical inference / George Casella, Roger L. Berger.-2nd ed.
Includes bibliographical references and indexes.
ISBN 0-534-24312-6
1. Mathematical statistics. 2. Probabilities.
I. Berger I Roger L.
Casella, George.
p. cm.
II .. Title.
QA216.C31 2001
519.5-dc21
2001025794
To Anne and Vicki
Duzbury titles 0/ related interest
Daniel, Applied Nonparametric Statistics 2nd
Derr, Statistical Consulting: A Guide to Effective Communication
Durrett, Probability: Theory and Examples 2nd
Graybill, Theory and Application of the Linear Model
Johnson, Applied Multivariate Methods for Data Analysts
Kuehl, Design of Experiments: Statistical Principles of Research Design and Analysis 2nd
Larsen, Marx, & Cooil, Statistics for Applied Problem Solving and Decision Making
Lohr, Sampling: Design and Analysis
Lunneborg, Data Analysis by Resampling: Concepts and Applications
Minh, Applied Probability Models
Minitab Inc., MINITABTM Student Version 12 for Windows
Myers, Classical and Modern Regression with Applications 2nd
Newton & Harvill, Stat Concepts: A Visual Tour of Statistical Ideas
Ramsey & Schafer, The Statistical Sleuth 2nd
SAS Institute Inc., JMP-IN: Statistical Discovery Software
Savage, INSIGHT: Business Analysis Software for Microsojt® Excel
Scheaffer, Mendenhall, & Ott, Elementary Survey Sampling 5th
Shapiro, Modeling the Supply Chain
Winston, Simu.lation Modeling Usin9..,(fRI~K·' , .
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Preface to the Second Edition
Although Sir Arthur Conan Doyle is responsible for most of the quotes in this book,
perhaps the best description of the life of this book can be attributed to the Grateful
Dead sentiment, "What a long, strange trip it's been."
Plans for the second edition started about six years ago, and for a long time we
struggled with questions about what to add and what to delete. Thankfully, as time
passed, the answers became clearer as the flow of the discipline of statistics became
clearer. We see tbe trend moving away from elegant proofs of special cases to algo
rithmic solutions of more complex and practical cases. This does not undermine the
importance of mathematics and rigor; indeed, we have found that these have become
more important. But the manner in which they are applied is changing.
For those familiar with the first edition, we can summarize the changes succinctly
as follows. Discussion of asymptotic methods has been greatly expanded into its own
chapter. There is more emphasis on computing and simulation (see Section 5.5 and
the computer algebra Appendix); coverage of the more applicable techniques has
been expanded or added (for example, bootstrapping, the EM algorithm, p-values,
logistic and robust regression); and there are many new Miscellanea and Exercises.
We have de-emphasized the more specialized theoretical topics, such as equivariance
and decision theory, and have restructured some material in Chapters 3-11 for clarity.
There are two things that we want to note. First, with respect to computer algebra
programs, although we believe that they are becoming increasingly valuable tools,
we did not want to force them on the instructor who does not share that belief.
Thus, the treatment is "unobtrusive" in that it appears only in an appendix, with
some hints throughout the book where it may be useful. Second, we have changed
the numbering system to one that facilitates finding things. Now theorems, lemmas,
examples, and definitions are numbered together; for example, Definition 7.2.4 is
followed by Example 7.2.5 and Theorem 10.1.3 precedes Example 10.1.4.
The first four chapters have received only minor changes. We reordered some ma
terial (in particular, the inequalities and identities have been split), added some new
examples and exercises, and did some general updating. Chapter 5 has also been re
ordered, witb the convergence section being moved further back, and a new section on
generating random variables added. The previous coverage of invariance, which was
in Chapters 7-9 of the first edition, has been greatly reduced and incorporated int.o
Chapter 6, which otherwise has received only minor editing (mostly the addition of
new exercises). Chapter 7 has been expanded and updated, and includes a new section
on the EM algorithm. Chapter 8 has also received minor editing and updating, and
.-MIJIA\'¥ )""'111
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PREFACE TO THE SECOND EDITION
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\: '~~s~ ~~e\\T" sjl~~?'4 p-values. In Chapter 9 we now put more emphasis on pivoting
(h8.vJn~~~hat "guaranteeing an interval" was merely "pivoting the cdf"). Also,
the rrrateihirihat was in Chapter 10 of the first edition (decision theory) has been re
duced, and small sections on loss function optimality of point estimation, hypothesis
testing, and interval estimation have been added to the appropriate chapters.
Chapter 10 is entirely new and attempts to layout the fundamentals of large sample
inference, including the delta method, consistency and asymptotic normality, boot
strapping, robust estimators, score tests, etc. Chapter 11 is classic oneway ANOVA
and linear regression (which was covered in two different chapters in the first edi
tion). Unfortunately, coverage of randomized block designs has been eliminated for
space reasons. Chapter 12 covers regression with errors-in-variables and contains new
material on robust and logistic regression.
After teaching from the first edition for a number of years, we know (approximately)
what can be covered in a one-year course. From the second edition, it should be
possible to cover the following in one year:
Chapter 1: Sections 1-7
Chapter 2: Sections 1-3
Chapter 3: Sections 1-6
Chapter 4: Sections 1-7
Chapter 5: Sections 1-6
Chapter 6: Sections 1-3
Chapter 7: Sections 1-3
Chapter 8: Sections 1-3
Chapter 9: Sections 1-3
Chapter 10: Sections 1, 3, 4
Classes that begin the course with some probability background can cover more ma
terial from the later chapters.
Finally, it is almost impossible to thank all of the people who have contributed in
some way to making the second edition a reality (and help us correct the mistakes in
the first edition). To all of our students, friends, and colleagues who took the time to
send us a note or an e-mail, we thank you. A number of people made key suggestions
that led to substantial changes in presentation. Sometimes these suggestions were just
short notes or comments, and some were longer reviews. Some were so long ago that
their authors may have forgotten, but we haven't. So thanks to Arthur Cohen, Sir
David Cox, Steve Samuels, Rob Strawderman and Tom Wehrly. We also owe much to
Jay Beder, who has sent us numerous comments and suggestions over the years and
possibly knows the first edition better than we do, and to Michael Perlman and his
class, who are sending comments and corrections even as we write this.
This book has seen a number of editors. We thank Alex Kugashev, who in the
mid-1990s first suggested doing a second edition, and our editor, Carolyn Crockett,
who constantly encouraged us. Perhaps the one person (other than us) who is most
responsible for this book is our first editor, John Kimmel, who encouraged, published,
and marketed the first edition. Thanks, John.
George Casella
Roger L. Berger
Preface to the First Edition
When someone discovers that you are writing a textbook, one (or both) of two ques
tions will be asked. The first is "Why are you writing a book?" and the second is
"How is your book different from what's out there?" The first question is fairly easy
to answer. You are writing a book because you are not entirely satisfied with the
available texts. The second question is harder to answer. The answer can't be put
in a few sentences so, in order not to bore your audience (who may be asking the
question only out of politeness), you try to say something quick and witty. It usually
doesn't work.
The purpose of this book is to build theoretical statistics (as different from mathe
matical statistics) from the first principles of probability theory. Logical development,
proofs, ideas, themes, etc., evolve through statistical arguments. Thus, starting from
the basics of probability, we develop the theory of statistical inference using tech
niques, definitions, and concepts that are statistical and are natural extensions and
consequences of previous concepts. When this endeavor was started, we were not sure
how well it would work. The final judgment of our success is, of course, left to the
reader.
The book is intended for first-year graduate students majoring in statistics or in
a field where a statistics concentration is desirable. The prerequisite is one year of
calculus. (Some familiarity with matrix manipulations would be useful, but is not
essential.) The book can be used for a two-semester, or three-quarter, introductory
course in statistics.
The first four chapters cover basics of probability theory and introduce many fun
damentals that are later necessary. Chapters 5 and 6 are the first statistical chapters.
Chapter 5 is transitional (between probability and statistics) and can be the starting
point for a course in statistical theory for students with some probability background.
Chapter 6 is somewhat unique, detailing three statistical principles (sufficiency, like
lihood, and invariance) and showing how these principles are important in modeling
data. Not all instructors will cover this chapter in detail, although we strongly recom
mend spending some time here. In particular, the likelihood and invariance principles
are treated in detail. Along with the sufficiency principle, these principles, and the
thinking behind them, are fundamental to total statistical understanding.
Chapters 7-9 represent the central core of statistical inference, estimation (point
and interval) and hypothesis testing. A major feature of these chapters is the division
into methods of finding appropriate statistical techniques and methods of evaluating
these techniques. Finding and evaluating are of interest to both the theorist and the
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