Lecture Notes in Statistics

1. Nonparametric Regression and Change-Point Problems.- 1.1. Introduction.- 1.2. The nonparametric regression problem.- 1.3. Kernel estimators.- 1.4. Locally-polynomial estimators.- 1.5. Piecewise-polynomial estimators.- 1.6. Bias and variance of the estimators.- 1.7. Criteria for comparing the nonparametric estimators.- 1.8. Rates of the uniform and L1- convergence.- 1.9. The change-point problem.- 2. Minimax Lower Bounds.- 2.1. General statistical model and minimax rates of convergence.- 2.2. The basic idea.- 2.3. Distances between distributions.- 2.4. Examples.- 2.5. The main theorem on lower bounds.- 2.6. Assouad's lemma.- 2.7. Examples: uniform and integral metrics.- 2.8. Arbitrary design.- 3. The Problem of Edge and Image Estimation.- 3.1. Introduction.- 3.2. Assumptions and notation.- 3.3. Lower bounds on the accuracy of estimates.- 4. Optimal Image and Edge Estimation for Boundary Fragments.- 4.1. Optimal edge estimation.- 4.2. Preliminary lemmas.- 4.3. Proof of Theorem 4.1.1.- 4.4. Optimal image estimation.- 4.5. Proof of Theorem 4.4.5.- 5. Generalizations and Extensions.- 5.1. High-dimensional boundary fragments. Non-Gaussian noise.- 5.2. General domains in high dimensions: a simple and rough estimator.- 5.3. Optimal estimators for general domains in two dimensions.- 5.4. Dudley's classes of domains.- 5.5. Maximum likelihood estimation on ?-net.- 5.6. Optimal edge estimators for Dudley's classes.- 5.7. On calculation of optimal edge estimators for general domains.- 6. Image Reconstruction Under Restrictions on Estimates.- 6.1. Naive linewise processing.- 6.2. Modified linewise procedure.- 6.3. Proofs.- 6.4. Linear image estimators.- 7. Estimation of Support of a Density.- 7.1. Problem statement.- 7.2. A simple and rough support estimator.- 7.3. Minimax lower bounds for support estimation.- 7.4. Optimal support estimation for boundary fragments.- 7.5. Optimal support estimation for convex domains and for Dudley's classes.- 8. Estimation of The Domain's Area.- 8.1. Preliminary discussion.- 8.2. Domain's area estimation in continuous parametric models.- 8.3. Theorem on the lower bound.- 8.4. Optimal estimator for the domain's area.- 8.5. Generalizations and extensions.- 8.6. Functionals of support of a density.- 9. Image Estimation from Indirect Observations.- 9.1. The blurred image model.- 9.2. High-dimensional blurred image models.- 9.3. Upper bounds in non-regular case.- 9.4. The stochastic problem of tomography.- 9.5. Minimax rates of convergence.- References.- Author Index.