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AuthorIngster, Yu. I. author
TitleNonparametric Goodness-of-Fit Testing Under Gaussian Models [electronic resource] / by Yu. I. Ingster, Irina A. Suslina
ImprintNew York, NY : Springer New York : Imprint: Springer, 2003
Connect tohttp://dx.doi.org/10.1007/978-0-387-21580-8
Descript XIV, 457 p. 1 illus. online resource

CONTENT

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 6.1โ{128}{147}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


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