zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Minimum complexity regression estimation with weakly dependent observations. (English) Zbl 0868.62015

Summary: The minimum complexity regression estimation framework, due to A. R. Barron [Nonparametric functional estimation and related topics, NATO ASI Ser., Ser. C 335, 561-576 (1991; Zbl 0739.62001)] is a general data-driven methodology for estimating a regression function from a given list of parametric models using independent and identically distributed (i.i.d.) observations. We extend Barron’s regression estimation framework to m-dependent observations and to strongly mixing observations.

In particular, we propose abstract minimum complexity regression estimators for dependent observations, which may be adapted to a particular list of parametric models, and establish upper bounds on the statistical risks of the proposed estimators in terms of certain deterministic indices of resolvability. Assuming that the regression function satisfies a certain Fourier-transform-type representation, we examine minimum complexity regression estimators adapted to a list of parametric models based on neural networks and, by using the upper bounds for the abstract estimators, we establish rates of convergence for the statistical risks of these estimators. Also, as a key tool, we extend the classical Bernstein inequality from i.i.d. random variables to m-dependent processes and to strongly mixing processes.

MSC:
62B10Statistical information theory
41A17Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
62M10Time series, auto-correlation, regression, etc. (statistics)