Foundations of Modern Statistics: Festschrift in Honor of Vladimir Spokoiny, Berlin, Germany, November 6–8, 2019, Moscow, Russia, November 30, 2019 ... Proceedings in Mathematics & Statistics, 425)

Description

This book contains contributions from the participants of the international conference “Foundations of Modern Statistics” which took place at Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, during November 6–8, 2019, and at Higher School of Economics (HSE University), Moscow, during November 30, 2019. The events were organized in honor of Professor Vladimir Spokoiny on the occasion of his 60th birthday. Vladimir Spokoiny has pioneered the field of adaptive statistical inference and contributed to a variety of its applications. His more than 30 years of research in the field of mathematical statistics had a great influence on the development of the mathematical theory of statistics to its present state. It has inspired many young researchers to start their research in this exciting field of mathematics.  The papers contained in this book reflect the broad field of interests of Vladimir Spokoiny: optimal rates and non-asymptotic bounds in nonparametrics, Bayes approaches from a frequentist point of view, optimization, signal processing, and statistical theory motivated by models in applied fields. Materials prepared by famous scientists contain original scientific results, which makes the publication valuable for researchers working in these fields. The book concludes by a conversation of Vladimir Spokoiny with Markus Reiβ and Enno Mammen. This interview gives some background on the life of Vladimir Spokoiny and his many scientific interests and motivations.   Preface Contents Optimal Rates and Non-asymptotic Bounds in Nonparametrics Adaptive Denoising of Signals with Local Shift-Invariant Structure 1 Introduction 2 Problem Description 2.1 Notation 2.2 Problem Statement 3 Oracle Inequalities for the ell2-Loss of Adaptive Estimators 3.1 Adaptive Signal Interpolation 3.2 Adaptive Signal Filtering 4 Risk Bounds for Adaptive Recovery Under ASI 4.1 Risk Bounds for Adaptive Signal Interpolation 4.2 Risk Bounds for Adaptive Signal Filtering 4.3 Harmonic Oscillation Denoising References Goodness-of-Fit Testing for Hölder Continuous Densities Under Local Differential Privacy 1 Introduction 2 Problem Statement 3 Non-interactive Privacy Mechanisms 3.1 Upper Bound in the Non-interactive Scenario 3.2 Lower Bound in the Non-interactive Scenario 4 Interactive Privacy Mechanisms 4.1 Upper Bound in the Interactive Scenario 4.2 Lower Bound in the Interactive Scenario 5 Examples A Proofs of Sect. 3 A.1 Proof of Proposition 3.2 A.2 Proof of Theorem 3.4 A.3 Proof of Lemma 3.7 B Proofs of Sect. 4 B.1 Proof of Proposition 4.1 B.2 Analysis of the Mean and Variance of the Statistic DB B.3 Proof of Theorem 4.3 B.4 Proof of Theorem 4.4 C Proofs of Sect. 5 C.1 Example 5.2 C.2 Example 5.3 C.3 Example 5.4 C.4 Example 5.5 C.5 Example 5.6 C.6 Example 5.7 C.7 Example 5.8 References Nonasymptotic One- and Two-Sample Tests in High Dimension with Unknown Covariance Structure 1 Introduction 1.1 Relation to White Noise Model in Nonparametric Statistics 1.2 Relation to ``Modern'' and High-Dimensional Statistics 1.3 Relation to Machine Learning and Kernel Mean Embeddings of Distributions 1.4 Overview of Contributions 1.5 Organization of the Paper 2 Main Results 2.1 A General Result to Upper Bound Separation Rates 2.2 Concentration Properties of the Test Statistic 2.3 Quantile Estimation 2.4 Concluding Remarks 3 Proofs 3.1 Proof of Theorem 3 3.2 Proof of Propositions 6 and 9 3.3 Proof of Theorem 7 3.4 Proof of Theorem 8 3.5 Proof of Propositions 10 and 11 3.6 Proof of Propositions 12 and 13 3.7 Additional Proofs References The Lasso with Structured Design and Entropy of (Absolute) Convex Hulls 1 Introduction 1.1 Organization of the Paper 2 Definitions 3 Bounds for the Empirical Process Using Approximation Numbers 4 Bounds for the Lasso Using Approximation Numbers 5 Some Entropy Results From the Literature 6 Explicit Entropy Bounds Using δ-Approximation Numbers 7 Bounds Using Covering Numbers Illustrated: One-Hidden-Layer Neural Networks 7.1 Definitions and Results 7.2 Simulation 8 Examples of Approximation Numbers 8.1 Second Order Discrete Derivatives 8.2 kth Order Discrete Derivatives 8.3 Higher-Dimensional Extensions 8.4 Entropy of the Class of Distribution Functions 9 Conclusion 10 Technical Proofs 10.1 Proof of Lemma 1 10.2 Proof of Lemma 2 10.3 Proof of Theorem 5 10.4 Proof of Lemma 3 10.5 Proof of Lemma 5 References Local Linear Smoothing in Additive Models as Data Projection 1 Introduction 2 Local Linear Smoothing In Additive Models 3 Existence and Uniqueness of the Estimator, Convergence of the Algorithm 4 Asymptotic Properties of the Estimator References A Multivariate CLT for Weighted Sums with Rate of Convergence of Order O(1/n) 1 Introduction and Main Result 2 Notation and Auxiliary Results 3 Proof of the Main Theorem References Estimation of Matrices and Subspaces Rate of Convergence for Sparse Sample Covariance Matrices 1 Introduction 2 Main Results 3 The Stieltjes Transforms Proximity 4 The Proof of Theorem 1 4.1 Truncation 4.2 The Proof of Theorem 5 The Proof of Theorem 2 5.1 Estimation of Resolvent Diagonal Elements 5.2 Estimation of Tn 6 Appendix References Van Trees Inequality, Group Equivariance, and Estimation of Principal Subspaces 1 Introduction 2 A Van Trees Inequality for the Estimation of Principal Subspaces 3 Proof of Proposition 1 3.1 Reduction to a Pointwise Risk 3.2 A Pointwise Cramér-Rao Inequality for Equivariant Estimators 4 Applications 4.1 PCA and the Subspace Distance 4.2 PCA and the Excess Risk 4.3 Low-Rank Matrix Denoising 5 Proofs for Sect.4 5.1 Specialization to Principal Subspaces 5.2 A Simple Optimization Problem 5.3 End of Proofs of the Consequences References Sparse Constrained Projection Approximation Subspace Tracking 1 Introduction 1.1 Main Setup 2 Methods 2.1 CPAST 2.2 Sparse CPAST 3 Error Bounds for CPAST and SCPAST 4 Numerical Results 5 Outlines of the Proofs 6 Conclusions 7 Proofs of Results in Section5 8 Concentration of the Spectral Norm of the Perturbation References Bernstein–von Mises Theorem and Misspecified Models: A Review 1 Introduction 2 Frequentist Results for Misspecified Models 2.1 Probability Model 2.2 Best Parameter 2.3 Regular Models 2.4 Nonasymptotic LAN Condition 2.5 Optimal Variance for Unbiased Estimators 3 Bernstein–von Mises Theorem for Correctly Specified Models 4 Bernstein–von Mises Theorem and Model Misspecification 4.1 Bayesian Inference Under Model Misspecification 4.2 Concentration 4.3 Bernstein–von Mises—Type Results Under Model Misspecification 4.4 Example: Misspecified Linear Model 5 ``Optimising'' Bayesian Inference Under Model Misspecification 5.1 Asymptotic Risk of Parameter Estimation Under a Misspecified Model 5.2 Composite Likelihoods 5.3 Generalised (Gibbs) Posterior Distribution 5.4 Nonparametric Model for Uncertainty in p0 and Bootstrap Posterior 5.5 Curvature Adjustment 6 Discussion and Open Questions References On Accuracy of Gaussian Approximation in Bayesian Semiparametric Problems 1 Introduction 1.1 Problem Statement 1.2 Related Work 2 Sieve Approach 3 Semiparametric Bernstein-von Mises Theorem 3.1 Parametric Estimation: Main Definitions 3.2 Conditions 3.3 Posterior Contraction 3.4 Gaussian Approximation of Posterior Distribution 3.5 Critical Dimension and Examples 4 Tools 4.1 Some Inequalities for Normal Distribution 4.2 Linear Approximation of Log-likelihood Gradient and Other Tools 5 Proofs of Main Results 5.1 Proof of Corollary 7 5.2 Proof of Corollary 8 5.3 Proof of Theorem 1 5.4 Proof of Theorem 2 5.5 Proof of Theorem 3 5.6 Proof of Theorem 4 5.7 Proof of Corollary 3 5.8 Proof of Theorem 6 5.9 Proof of Theorem 7 References Statistical Theory Motivated by Applications An Alternative to Synthetic Control for Models with Many Covariates Under Sparsity 1 Introduction 2 Covariate Balancing Weights and Double Robustness 3 A Parametric Alternative to Synthetic Control 3.1 Estimation With Low-Dimensional Covariates 3.2 Estimation With High-Dimensional Covariates 3.3 Asymptotic Properties 4 Monte Carlo Simulations 5 Empirical Applications 5.1 Job Training Program 5.2 California Tobacco Control Program 6 Conclusion 7 Algorithm for Feasible Penalty Loadings 8 Proofs 9 Auxiliary Lemmas References Simple Adaptive Estimation of Quadratic Functionals in Nonparametric IV Models 1 Introduction 2 Minimax Optimal Quadratic Functional Estimation 2.1 Preliminaries and Notation 2.2 A Leave-one-out, Sieve NPIV Estimator 2.3 Rate of Convergence 3 Rate Adaptive Estimation 4 Conclusion and Extensions 5 Proofs of Results in Section 2 6 Proofs of Results in Section 3 7 Supplementary Lemmas References A Minimax Testing Perspective on Spatial Statistical Resolution in Microscopy 1 Introduction 2 Theory 2.1 Assumptions 2.2 Results 2.3 Physical Implications 3 Simulations 4 Proofs 4.1 Most difficult Setup for Even psfs in the CLT Regime References Optimization Unifying Framework for Accelerated Randomized Methods in Convex Optimization 1 Introduction 1.1 Related Work 1.2 Our Approach and Contributions 2 Preliminaries 2.1 Notation 2.2 Problem Statement and Assumptions 3 Unified Accelerated Randomized Method 4 Extension for Strongly Convex Functions 5 Examples of Applications 5.1 Accelerated Random Directional Search 5.2 Accelerated Random Coordinate Descent 5.3 Accelerated Random Block-Coordinate Descent 5.4 Accelerated Random Derivative-Free Directional Search 5.5 Accelerated Random Derivative-Free Coordinate Descent 5.6 Accelerated Random Derivative-Free Block-Coordinate Descent 5.7 Accelerated Random Derivative-Free Block-Coordinate Descent with Random Approximations for Block Derivatives 6 Model Generality in a Non-accelerated Random Block-Coordinate Descent 7 Conclusion References Surrogate Models for Optimization of Dynamical Systems 1 Introduction 2 Literature Review 3 Mathematical Framework 3.1 Optimal Control Problem for Dynamical Systems 3.2 Surrogate Models for Optimization Problems 4 Enhanced Surrogate Models 4.1 Iterative Algorithm 5 Application of POD-RBF Procedure on Dynamical Systems 5.1 Model 1: Science Policy 5.2 Model 2: Population Dynamics 5.3 Model 3: Quality Control in Production and Process Management 6 Conclusion 6.1 Limitations and Future Work References Appendix Interview with Vladimir Spokoiny on 29/01/21 by E. Mammen and M. Reiß