Estimation of a function with discontinuities via local polynomial fit with an adaptive window choice
- Spokoiny, Vladimir
2010 Mathematics Subject Classification
- 62G07 62G20
- change-point, local polynomial fit local structure, nonparametric regression, pointwise adaptive estimation
New method of adaptive estimation of a regression function is proposed. The resulting estimator achieves near optimal rate of estimation in the classical sense of mean integrated squared error. At the same time, the estimator is shown to be very sensitive to discontinuities or change-points of the underlying function ƒ or its derivatives. For instance, in the case of a jump of a regression function, beyond the interval of length (in order) n-1 log n around change-points the quality of estimation is essentially the same as if the location of this jump 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 non-asymptotic way and can be therefore applied for an arbitrary sample size.
- Ann. Statist., 26 (1998), No. 4, pp. 1356-1378