Estimation of a Function with Discontinuities via Local Polynomial Fit with an Adaptive Window Choice

We propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function f or its derivatives. For instan...

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Bibliographic Details
Published in:The Annals of statistics 1998-08, Vol.26 (4), p.1356-1378
Main Author: Spokoiny, V. G.
Format: Article
Language:English
Subjects:
62G07
62G20
Analytical estimating
Change Points and Discontinuities
Change-point
Estimate reliability
Estimation methods
Estimators
Exact sciences and technology
Linear inference, regression
local polynomial fit
local structure
Mathematical functions
Mathematical vectors
Mathematics
Matrices
Nonparametric inference
nonparametric regression
Point estimators
pointwise adaptive estimation
Polynomials
Probability and statistics
Sciences and techniques of general use
Statistical variance
Statistics
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Summary:We propose a method of adaptive estimation of a regression function which is near optimal in the classical sense of the mean integrated error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function f or its derivatives. For instance, in the case of a jump of a regression function, beyond the intervals of length (in order) n-1log n around change-points the quality of estimation is essentially the same as if locations of jumps were known. The method is fully adaptive and no assumptions are imposed on the design, number and size of jumps. The results are formulated in a nonasymptotic way and can therefore be applied for an arbitrary sample size.
ISSN:0090-5364
2168-8966
DOI:10.1214/aos/1024691246