Nonparametric Goodness-of-Fit Testing Under Gaussian Models
Main Authors: | , |
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Format: | eBook |
Language: | English |
Published: |
New York, NY
Springer New York
2003, 2003
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Edition: | 1st ed. 2003 |
Series: | Lecture Notes in Statistics
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Subjects: | |
Online Access: | |
Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 1 Introduction
- 1.1 Tests
- 1.2 One-Dimensional Parameter
- 1.3 Multidimensional Parameter
- 1.4 Infinite-Dimensional Parameter
- 1.5 Problems of the Study and Main Results
- 1.6 Methods of the Study
- 1.7 Structure of the Book
- 2 An Overview
- 2.1 Models
- 2.2 Hypothesis Testing Problem
- 2.3 Bayesian Approach in Hypothesis Testing
- 2.4 Minimax Approach in Hypothesis Testing
- 2.5 Asymptotics in Hypothesis Testing
- 2.6 Minimax Distinguishability in Goodness-of-Fit Problems
- 2.7 Norms and Wavelet Transform
- 2.8 Short Overview of Minimax Estimation
- 2.9 Constraints of Interest
- 2.10 Rates in Estimation and in Hypothesis Testing
- 3 Minimax Distinguishability
- 3.1 Minimax Properties of Test Families
- 3.2 Asymptotic Minimaxity for Square Norms
- 3.3 Bayesian Approach under a Gaussian Model
- 3.4 Triviality and Classical Asymptotics
- 3.5 Distinguishability Conditions for Smooth Signals
- 4 Sharp Asymptotics. I
- 4.1 Tests Based on Linear Statistics and Convex Alternatives
- 4.2 Two-Sided Constraints for the Positive Alternatives, p ? 1, q ? p
- 4.3 Sharp Asymptotics of Gaussian Type: Product Priors
- 4.4 Sharp Asymptotics: Asymptotic Degeneracy
- 5 Sharp Asymptotics. II
- 5.1 Tests Based on Log-Likelihood Statistics and Thresholding
- 5.2 Extreme Problem in the Space of Sequences of Measures
- 5.3 Separation of the Problem
- 5.4 Solution of One-Dimensional Problems
- 5.5 Sharp Asymptotics for ln-Balls
- 6 Gaussian Asymptotics for Power and Besov Norms
- 6.1 Extreme Problems
- 6.2 Principal Types of Gaussian Asymptotics
- 6.3 Frontier Log-Types of Gaussian Asymptotics
- 6.4 Graphical Presentation
- 6.5 Remarks on the Proofs of Theorems 6.1–6.4
- 6.6 Proof of Theorems 6.1 and 6.3 for p ? 2, q ? p, and p = q
- 6.7 Extreme Problem for Power Norms: p ? q.-6.8 Properties of the Extreme Sequences for Power Norms
- 6.9 Extreme Problem for Besov Norms
- A.4.7 Proof of Proposition 6.3
- A.5 Proof of Lemma 7.4
- A.6 Proofs of Lemmas 8.2, 8.3, 8.4, 8.6
- References
- Parameter and Function Index
- 7 Adaptation for Power and Besov Norms
- 7.1 Adaptive Setting
- 7.2 Lower Bounds
- 7.3 Upper Bounds for Power Norms
- 7.4 Upper Bounds for Besov Norms
- 8 High-Dimensional Signal Detection
- 8.1 The Bayesian Signal Detection Problem
- 8.2 Multichannel Signal Detection Problems
- 8.3 Minimax Signal Detection for ln-Balls
- 8.4 Proof of Upper Bounds
- 8.5 Testing a Hypothesis which Is Close to a Simple Hypothesis
- A Appendix
- A.1 Proof of Proposition 2.16
- A.2 Proof of Proposition 5.3
- A.2.1 Properties of Statistics under Alternatives
- A.2.2 Evaluations of Type II Errors
- A.3 Study of the Extreme Problem for Power Norms
- A.3.1 Solution of the System (6.86), (6.87)
- A.3.4 Solution of the Extreme Problem (6.88)
- A.3.8 Proofs of Propositions 6.1, 6.2
- A.4 Study of the Extreme Problem for Besov Norms
- A.4.1 Solution of the System (6.110), (6.111)
- A.4.2 Solution of the Extreme Problem (6.112)
- A.4.5 Upper Bounds
- A.4.6 Lower Bounds