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E-raamat: Mathematical Statistics

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Mathematical StatisticsSuurem pilt
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This book is designed to bridge the gap between traditional textbooks in statistics and more advanced books that include the sophisticated nonparametric techniques. It covers topics in parametric and nonparametric large-sample estimation theory. The exposition is based on a collection of relatively simple statistical models. It gives a thorough mathematical analysis for each of them with all the rigorous proofs and explanations. The book also includes a number of helpful exercises. Prerequisites for the book include senior undergraduate/beginning graduate-level courses in probability and statistics.

Arvustused

This is a well written book and should be of great interest to advanced graduate students/researchers in mathematical statistics. The material is presented with great clarity by using simple models as opposed to complex ones. ... Overall it should be of great value to advanced graduate students and researchers in theoretical statistics. The book can be recommended for libraries on campuses with a graduate program in statistics." - Mathematical Reviews

Preface ix
Part 1 Parametric Models
Chapter 1 The Fisher Efficiency
3(8)
1.1 Statistical Experiment
3(3)
1.2 The Fisher Information
6(1)
1.3 The Cramer-Rao Lower Bound
7(1)
1.4 Efficiency of Estimators
8(3)
Exercises
9(2)
Chapter 2 The Bayes and Minimax Estimators
11(10)
2.1 Pitfalls of the Fisher Efficiency
11(2)
2.2 The Bayes Estimator
13(3)
2.3 Minimax Estimator. Connection Between Estimators
16(2)
2.4 Limit of the Bayes Estimator and Minimaxity
18(3)
Exercises
19(2)
Chapter 3 Asymptotic Minimaxity
21(22)
3.1 The Hodges Example
21(1)
3.2 Asymptotic Minimax Lower Bound
22(4)
3.3 Sharp Lower Bound. Normal Observations
26(2)
3.4 Local Asymptotic Normality (LAN)
28(3)
3.5 The Hellinger Distance
31(2)
3.6 Maximum Likelihood Estimator
33(2)
3.7 Proofs of Technical Lemmas
35(8)
Exercises
40(3)
Chapter 4 Some Irregular Statistical Experiments
43(8)
4.1 Irregular Models: Two Examples
43(1)
4.2 Criterion for Existence of the Fisher Information
44(1)
4.3 Asymptotically Exponential Statistical Experiment
45(2)
4.4 Minimax Rate of Convergence
47(1)
4.5 Sharp Lower Bound
47(4)
Exercises
49(2)
Chapter 5 Change-Point Problem
51(14)
5.1 Model of Normal Observations
51(3)
5.2 Maximum Likelihood Estimator of Change Point
54(2)
5.3 Minimax Limiting Constant
56(1)
5.4 Model of Non-Gaussian Observations
57(2)
5.5 Proofs of Lemmas
59(6)
Exercises
62(3)
Chapter 6 Sequential Estimators
65(20)
6.1 The Markov Stopping Time
65(4)
6.2 Change-Point Problem. Rate of Detection
69(4)
6.3 Minimax Limit in the Detection Problem
73(2)
6.4 Sequential Estimation in the Autoregressive Model
75(10)
Exercises
83(2)
Chapter 7 Linear Parametric Regression
85(16)
7.1 Definitions and Notations
85(2)
7.2 Least-Squares Estimator
87(2)
7.3 Properties of the Least-Squares Estimator
89(4)
7.4 Asymptotic Analysis of the Least-Squares Estimator
93(8)
Exercises
96(5)
Part 2 Nonparametric Regression
Chapter 8 Estimation in Nonparametric Regression
101(14)
8.1 Setup and Notations
101(2)
8.2 Asymptotically Minimax Rate of Convergence. Definition
103(1)
8.3 Linear Estimator
104(2)
8.4 Smoothing Kernel Estimator
106(9)
Exercises
112(3)
Chapter 9 Local Polynomial Approximation of the Regression Function
115(16)
9.1 Preliminary Results and Definition
115(4)
9.2 Polynomial Approximation and Regularity of Design
119(3)
9.3 Asymptotically Minimax Lower Bound
122(4)
9.4 Proofs of Auxiliary Results
126(5)
Exercises
130(1)
Chapter 10 Estimation of Regression in Global Norms
131(20)
10.1 Regressogram
131(2)
10.2 Integral Z/2-Norm Risk for the Regressogram
133(3)
10.3 Estimation in the Sup-Norm
136(2)
10.4 Projection on Span-Space and Discrete MISE
138(3)
10.5 Orthogonal Series Regression Estimator
141(10)
Exercises
148(3)
Chapter 11 Estimation by Splines
151(16)
11.1 In Search of Smooth Approximation
151(1)
11.2 Standard B-splines
152(3)
11.3 Shifted B-splines and Power Splines
155(3)
11.4 Estimation of Regression by Splines
158(3)
11.5 Proofs of Technical Lemmas
161(6)
Exercises
166(1)
Chapter 12 Asymptotic Optimality in Global Norms
167(18)
12.1 Lower Bound in the Sup-Norm
167(4)
12.2 Bound in L2-Norm. Assouad's Lemma
171(3)
12.3 General Lower Bound
174(3)
12.4 Examples and Extensions
177(8)
Exercises
182(3)
Part 3 Estimation in Nonparametric Models
Chapter 13 Estimation of Functionals
185(8)
13.1 Linear Integral Functionals
185(3)
13.2 Non-Linear Functionals
188(5)
Exercises
191(2)
Chapter 14 Dimension and Structure in Nonparametric Regression
193(18)
14.1 Multiple Regression Model
193(3)
14.2 Additive regression
196(3)
14.3 Single-Index Model
199(7)
14.4 Proofs of Technical Results
206(5)
Exercises
209(2)
Chapter 15 Adaptive Estimation
211(16)
15.1 Adaptive Rate at a Point. Lower Bound
211(4)
15.2 Adaptive Estimator in the Sup-Norm
215(3)
15.3 Adaptation in the Sequence Space
218(5)
15.4 Proofs of Lemmas
223(4)
Exercises
225(2)
Chapter 16 Testing of Nonparametric Hypotheses
227(12)
16.1 Basic Definitions
227(2)
16.2 Separation Rate in the Sup-Norm
229(2)
16.3 Sequence Space. Separation Rate in the L2-Norm
231(8)
Exercises
237(2)
Bibliography 239(2)
Index of Notation 241(2)
Index 243
Alexander Korostelev Wayne State University, Detroit, MI, USA

Olga Korosteleva, California State University, Long Beach, CA, USA
Lisainfo e-raamatute kohta Lisainfo e-raamatute kohta
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