Nemirovski A. Topics in non-parametric statistics

Saint-Flour: XXVII Saint-Flour Summer School on Probability and Statistics, 1998. — 200 p.Estimating regression functions from Hölder balls
Introduction
Recovering a univariate Lipschitz continuous function
Extension: recovering functions from Hölder balls
Appendix: proof of the Fano inequality
Estimating regression functions from Sobolev balls
Lower bounds for the minimax risk
Upper bounds on the minimax risk
Appendix: Proofs of Theorems 2.1.1, 2.1.2
Spatial adaptive estimation on Sobolev balls
Spatial adaptive estimation: the goal
The estimate
Quality of estimation
Optimality index of the adaptive estimate
Estimating signals satisfying differential inequalities
The goal
Estimating solutions of homogeneous equations
Preliminaries
Estimating sequences
Discussion
From sequences to functions
Spatial adaptive estimate: preliminaries
Spatial adaptive estimate: construction and quality
``Frequency modulated signals''
Appendix: Proofs of Lemmas 4.2.1, 4.3.1
Proof of Lemma 4.2.1
Proof of Lemma 4.3.1
Aggregation of estimates, I
Motivation
The problem and the main result
Aggregation problem
The recovering routine
Main result
``Concentration''
Lower bound
Application: Recovering functions from Barron's class
Numerical example: nonparametric filtration
Aggregation of estimates, II
Gaussian white noise model of observations
Approximating the best linear combination of estimates
Application: aggregating projection estimates
Linear estimates
Aggregating projection estimates
The construction
Approximating the best of given estimates
Estimating functionals, I
The problem
Lower bounds and asymptotical efficiency
The case of once continuously differentiable functional
Whether condition (7.2.2) is sharp?
Increasing smoothness of F
The case of twice continuously differentiable functional
Concluding remarks
Estimating functionals, II
Preliminaries: estimating polynomials
Hilbert-Schmidt polynomials
Estimating Hilbert-Schmidt polynomials
Extension
From polynomials to smooth functionals
Measure concentration
The estimate