Mathematical Statistics,9780387986746

Mathematical Statistics

by
ISBN13: 9780387986746
ISBN10: 038798674X
Format: Hardcover
Pub. Date: 1999-03-01
Publisher(s): Springer Verlag
List Price: $89.95

Rent Textbook

Select for Price
Add to Cart
There was a problem. Please try again later.

Rent Digital

Rent Digital Options
Online:30 Days access
Downloadable:30 Days
$32.04
Online:60 Days access
Downloadable:60 Days
$42.72
Online:90 Days access
Downloadable:90 Days
$53.40
Online:120 Days access
Downloadable:120 Days
$64.08
Online:180 Days access
Downloadable:180 Days
$69.42
Online:1825 Days access
Downloadable:Lifetime Access
$106.80
*To support the delivery of the digital material to you, a digital delivery fee of $3.99 will be charged on each digital item.
$69.42*
Add to Cart

New Textbook

We're Sorry
Sold Out

Used Textbook

We're Sorry
Sold Out

Summary

Covers topics in statistical theory essential for graduate students preparing for work on a Ph.D. degree in statistics. DLC: Mathematical statistics.

Author Biography

Jun Shao is Professor of Statistics at the University of Wisconsin, Madison.

Table of Contents

Preface vii
Chapter 1. Probability Theory
1(60)
1.1 Probability Spaces and Random Elements
1(8)
1.1.1 Sigma-fields and measures
1(5)
1.1.2 Measurable functions and distributions
6(3)
1.2 Integration and Differentiation
9(8)
1.2.1 Integration
9(5)
1.2.2 Radon-Nikodym derivative
14(3)
1.3 Distributions and Their Characteristics
17(13)
1.3.1 Useful probability densities
17(8)
1.3.2 Moments and generating functions
25(5)
1.4 Conditional Expectations
30(8)
1.4.1 Conditional expectations
30(4)
1.4.2 Independence
34(2)
1.4.3 Conditional distributions
36(2)
1.5 Asymptotic Theorems
38(11)
1.5.1 Convergence modes and stochastic orders
38(4)
1.5.2 Convergence of transformations
42(3)
1.5.3 The law of large numbers
45(2)
1.5.4 The central limit theorem
47(2)
1.6 Exercises
49(12)
Chapter 2. Fundamentals of Statistics
61(66)
2.1 Populations, Samples, and Models
61(9)
2.1.1 Populations and samples
61(3)
2.1.2 Parametric and nonparametric models
64(2)
2.1.3 Exponential and location-scale families
66(4)
2.2 Statistics and Sufficiency
70(13)
2.2.1 Statistics and their distributions
70(3)
2.2.2 Sufficiency and minimal sufficiency
73(6)
2.2.3 Complete statistics
79(4)
2.3 Statistical Decision Theory
83(9)
2.3.1 Decision rules, loss functions, and risks
83(3)
2.3.2 Admissibility and optimality
86(6)
2.4 Statistical Inference
92(9)
2.4.1 Point estimators
92(3)
2.4.2 Hypothesis tests
95(4)
2.4.3 Confidence sets
99(2)
2.5 Asymptotic Criteria and Inference
101(11)
2.5.1 Consistency
102(3)
2.5.2 Asymptotic bias, variance, and mse
105(4)
2.5.3 Asymptotic inference
109(3)
2.6 Exercises
112(15)
Chapter 3. Unbiased Estimation
127(66)
3.1 The UMVUE
127(13)
3.1.1 Sufficient and complete statistics
128(4)
3.1.2 A necessary and sufficient condition
132(3)
3.1.3 Information inequality
135(3)
3.1.4 Asymptotic properties of UMVUE's
138(2)
3.2 U-Statistics
140(8)
3.2.1 Some examples
140(2)
3.2.2 Variances of U-statistics
142(2)
3.2.3 The projection method
144(4)
3.3 The LSE in Linear Models
148(13)
3.3.1 The LSE and estimability
148(4)
3.3.2 The UMVUE and BLUE
152(3)
3.3.3 Robustness of LSE's
155(4)
3.3.4 Asymptotic properties of LSE's
159(2)
3.4 Unbiased Estimators in Survey Problems
161(9)
3.4.1 UMVUE's of population totals
161(4)
3.4.2 Horvitz-Thompson estimators
165(5)
3.5 Asymptotically Unbiased Estimators
170(12)
3.5.1 Functions of unbiased estimators
170(3)
3.5.2 The method of moments
173(3)
3.5.3 V-statistics
176(3)
3.5.4 The weighted LSE
179(3)
3.6 Exercises
182(11)
Chapter 4. Estimation in Parametric Models
193(84)
4.1 Bayes Decisions and Estimators
193(20)
4.1.1 Bayes actions
193(5)
4.1.2 Empirical and hierarchical Bayes methods
198(3)
4.1.3 Bayes rules and estimators
201(6)
4.1.4 Markov chain Monte Carlo
207(6)
4.2 Invariance
213(10)
4.2.1 One-parameter location families
213(4)
4.2.2 One-parameter scale families
217(2)
4.2.3 General location-scale families
219(4)
4.3 Minimaxity and Admissibility
223(12)
4.3.1 Estimators with constant risks
223(4)
4.3.2 Results in one-parameter exponential families
227(2)
4.3.3 Simultaneous estimation and shrinkage estimators
229(6)
4.4 The Method of Maximum Likelihood
235(13)
4.4.1 The likelihood function and MLE's
235(6)
4.4.2 MLE's in generalized linear models
241(4)
4.4.3 Quasi-likelihoods and conditional likelihoods
245(3)
4.5 Asymptotically Efficient Estimation
248(13)
4.5.1 Asymptotic optimality
248(4)
4.5.2 Asymptotic efficiency of MLE's and RLE's
252(5)
4.5.3 Other asymptotically efficient estimators
257(4)
4.6 Exercises
261(16)
Chapter 5. Estimation in Nonparametric Models
277(68)
5.1 Distribution Estimators
277(14)
5.1.1 Empirical c.d.f.'s in i.i.d. cases
278(3)
5.1.2 Empirical likelihoods
281(7)
5.1.3 Density estimation
288(3)
5.2 Statistical Functionals
291(13)
5.2.1 Differentiability and asymptotic normality
291(5)
5.2.2 L-, M-, R-estimators and rank statistics
296(8)
5.3 Linear Functions of Order Statistics
304(8)
5.3.1 Sample quantiles
304(4)
5.3.2 Robustness and efficiency
308(3)
5.3.3 L-estimators in linear models
311(1)
5.4 Generalized Estimating Equations
312(13)
5.4.1 The GEE method and its relationship with others
313(4)
5.4.2 Consistency of GEE estimators
317(4)
5.4.3 Asymptotic normality of GEE estimators
321(4)
5.5 Variance Estimation
325(12)
5.5.1 The substitution method
326(3)
5.5.2 The jackknife
329(5)
5.5.3 The bootstrap
334(3)
5.6 Exercises
337(8)
Chapter 6. Hypothesis Tests
345(76)
6.1 UMP Tests
345(11)
6.1.1 The Neyman-Pearson lemma
346(3)
6.1.2 Monotone likelihood ratio
349(4)
6.1.3 UMP tests for two-sided hypotheses
353(3)
6.2 UMP Unbiased Tests
356(13)
6.2.1 Unbiasedness and similarity
356(2)
6.2.2 UMPU tests in exponential families
358(4)
6.2.3 UMPU tests in normal families
362(7)
6.3 UMP Invariant Tests
369(11)
6.3.1 Invariance and UMPI tests
369(5)
6.3.2 UMPI tests in normal linear models
374(6)
6.4 Tests in Parametric Models
380(14)
6.4.1 Likelihood ratio tests
380(3)
6.4.2 Asymptotic tests based on likelihoods
383(4)
6.4.3 X(2)-tests
387(5)
6.4.4 Bayes tests
392(2)
6.5 Tests in Nonparametric Models
394(12)
6.5.1 Sign, permutation, and rank tests
394(4)
6.5.2 Kolmogorov-Smirnov and Cramer-von Mises tests
398(3)
6.5.3 Empirical likelihood ratio tests
401(3)
6.5.4 Asymptotic tests
404(2)
6.6 Exercises
406(15)
Chapter 7. Confidence Sets
421(68)
7.1 Construction of Confidence Sets
421(13)
7.1.1 Pivotal quantities
421(6)
7.1.2 Inverting acceptance regions of tests
427(3)
7.1.3 The Bayesian approach
430(2)
7.1.4 Prediction sets
432(2)
7.2 Properties of Confidence Sets
434(11)
7.2.1 Lengths of confidence intervals
434(4)
7.2.2 UMA and UMAU confidence sets
438(3)
7.2.3 Randomized confidence sets
441(2)
7.2.4 Invariant confidence sets
443(2)
7.3 Asymptotic Confidence Sets
445(8)
7.3.1 Asymptotically pivotal quantities
445(2)
7.3.2 Confidence sets based on likelihoods
447(4)
7.3.3 Results for quantiles
451(2)
7.4 Bootstrap Confidence Sets
453(14)
7.4.1 Construction of bootstrap confidence intervals
453(4)
7.4.2 Asymptotic correctness and accuracy
457(6)
7.4.3 High-order accurate bootstrap confidence sets
463(4)
7.5 Simultaneous Confidence Intervals
467(8)
7.5.1 Bonferroni's method
468(1)
7.5.2 Scheffe's method in linear models
469(2)
7.5.3 Tukey's method in one-way ANOVA models
471(2)
7.5.4 Confidence bands for c.d.f.'s
473(2)
7.6 Exercises
475(14)
Appendix A. Abbreviations 489(2)
Appendix B. Notation 491(2)
References 493(12)
Author Index 505(4)
Subject Index 509

An electronic version of this book is available through VitalSource.

This book is viewable on PC, Mac, iPhone, iPad, iPod Touch, and most smartphones.

By purchasing, you will be able to view this book online, as well as download it, for the chosen number of days.

Digital License

You are licensing a digital product for a set duration. Durations are set forth in the product description, with "Lifetime" typically meaning five (5) years of online access and permanent download to a supported device. All licenses are non-transferable.

More details can be found here.

A downloadable version of this book is available through the eCampus Reader or compatible Adobe readers.

Applications are available on iOS, Android, PC, Mac, and Windows Mobile platforms.

Please view the compatibility matrix prior to purchase.