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Nonparametric Goodness-of-Fit Testing Under Gaussian Models

Bibliographic Details
Main Authors: Ingster, Yuri, Suslina, I.A. (Author)
Format: eBook
Language:English
Published: New York, NY Springer New York 2003, 2003
Edition:1st ed. 2003
Series:Lecture Notes in Statistics
Subjects:
Statistical Theory And Methods
Statistics 
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

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