Suslina / Ingster - Nonparametric Goodness-of-Fit Testing Under Gaussian Models

Nonparametric Goodness-of-Fit Testing Under Gaussian Models

Medium: Buch

ISBN: 978-0-387-95531-5

Verlag: Springer

Erscheinungstermin: 29.10.2002

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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.- 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.- 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.