×

zbMATH — the first resource for mathematics

Fully data-driven nonparametric variance estimators. (English) Zbl 0811.62047
Summary: We consider the problem of estimating the unknown variance function \(v\) in a nonparametric regression model. As a basis for our estimators we take estimated residuals which are based on a kernel estimator of the mean vector. Then we form with these residuals a kernel estimator of \(v\). Main emphasis is on a data-driven choice of the bandwidths involved in the procedure. It is shown that the risk of this estimator attains the uniform convergence rate in Sobolev classes for \(v\) under weak smoothness assumptions on the mean. Moreover, we prove that there is asymptotically no loss due to the estimation of the mean.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bunke O., Seminarbericht Nr. 89 pp 65– (1986)
[2] DOI: 10.1214/aos/1176345987 · Zbl 0571.62058
[3] Carroll R. J., Transformations and Weighting in Regression (1988) · Zbl 0666.62062
[4] DOI: 10.1007/BFb0098489
[5] Gasser T., J. Roy. Statist. Soc. 47 pp 238– (1985)
[6] Hall P., J. Roy. Statist. Soc. 51 pp 3– (1989)
[7] DOI: 10.1093/biomet/77.3.521 · Zbl 1377.62102
[8] DOI: 10.1093/biomet/77.2.415 · Zbl 0711.62035
[9] DOI: 10.1214/aos/1176349748 · Zbl 0594.62043
[10] DOI: 10.1214/aoms/1177697731 · Zbl 0193.47201
[11] Johnson N. L., Distributions in statistics, continuous univariate distributions – 1 (1970) · Zbl 0213.21101
[12] DOI: 10.1214/aos/1176350364 · Zbl 0632.62040
[13] Neumann M. H., Seminarbericht Nr. 109 (1990)
[14] Neumann M. H., Preprint Nr. 91–21 (1991)
[15] DOI: 10.1214/aos/1176345206 · Zbl 0451.62033
[16] DOI: 10.1214/aos/1176345969 · Zbl 0511.62048
[17] DOI: 10.1137/1105028
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.