Nonparametric Goodness-of-Fit Testing Under Gaussian Models

• 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.

From the reviews: "The book is self-contained, and the bibliography is very rich and in fact provides a comprehensive listing of references about minimax testing (something that heretofore had been missing from the field.) To get the best out of this book, the reader should be familiar with basic functional analysis, wavelet theory, and optimization for extreme problems...It is highly recommended to anyone who wants an introduction to hypothesis testing from the minimax approach-yet it is only a starting point, as Gaussian models are studied exclusively." Journal of the American Statistical Association, June 2004 "The book deals with nonparametric goodness-of-fit testing problems from the literature of the past twenty years. ... It is a theoretical book with mathematical results ... . The proofs of the theorems are very detailed and many details are in the appendix of more than one hundred pages." (N. D. C. Veraverbeke, Short Book Reviews, Vol. 24 (1), 2004) "The present book is devoted to a modern theory of nonparametric goodness-of-fit testing. ... The level of the book meets a quite high standard. The book will certainly be of interest to mathematical statisticians interested in the theory of nonparametric statistical interference, and also to specialists dealing with applied nonparametric statistical problems in signal detection and transmission, technical and medical diagnostics, and other fields." (Marie Huskova, Zentralblatt MATH, Vol. 1013, 2003)