×
Galtchouk, Leonid; Pergamenshchikov, Sergey

Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression. (English) Zbl 1293.62091

J. Korean Stat. Soc. 38, No. 4, 305-322 (2009).
Summary: The paper considers some asymptotic properties of the adaptive procedure proposed in the authors’ paper [“Adaptive nonparametric estimation in heteroscedastic regression models. I: Sharp non-asymptotic oracle inequalities”, J. Nonparametric Stat. 21, No. 1, 1–16 (2009), http://hal.archives-ouvertes.fr/hal-00269196] for estimating an unknown nonparametric regression. We show that the procedure is asymptotically efficient for quadratic risk, i.e. the asymptotic quadratic risk of the procedure coincides with the corresponding Pinsker constant provided the sharp lower bound for the quadratic risk over all possible estimators.

Cited in 17 Documents

MSC:

62G08 Nonparametric regression and quantile regression
62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference

Keywords:

asymptotic bounds; adaptive estimation; efficient estimation; heteroscedastic regression; nonparametric regression; Pinsker’s constant
PDF BibTeX XML Cite
\textit{L. Galtchouk} and \textit{S. Pergamenshchikov}, J. Korean Stat. Soc. 38, No. 4, 305--322 (2009; Zbl 1293.62091)
Full Text: DOI arXiv
  • logo of hbz
  • logo of WorldCat

References:

[1] Belomestny, D.; Reiss, M., Spectral calibration of exponential Lévy models, Finance and stochastics, 10, 4, 449-474, (2006) · Zbl 1126.91022
[2] Brua, J.-Y., Asymptotic efficient estimators for non-parametric heteroscedastic model, Statistical methodology, 6, 1, 47-60, (2009) · Zbl 1220.62039
[3] Efromovich, S., Sequential design and estimation in heteroscedastic nonparametric regression, Sequential analysis, 26, 3-25, (2007) · Zbl 1112.62080
[4] Efromovich, S.; Pinsker, M., Sharp-optimal and adaptive estimation for heteroscedastic nonparametric regression, Statistica sinica, 6, 925-942, (1996) · Zbl 0857.62037
[5] Galtchouk, L.; Pergamenshchikov, S., Asymptotically efficient estimates for nonparametric regression models, Statistics & probability letters, 76, 852-860, (2006) · Zbl 1089.62044
[6] Galtchouk, L., & Pergamenshchikov, S. (2005). Efficient adaptive nonparametric estimation in heteroscedastic regression models. Preprint of the Strasbourg Louis Pasteur University, IRMA, 2005/020. Available at: http://www.univ-rouen.fr/LMRS/Persopage/Pergamenchtchikov · Zbl 1293.62091
[7] Galtchouk, L., & Pergamenshchikov, S. (2007). Adaptive nonparametric estimation in heteroscedastic regression models. part 1. Sharp non-asymptotic oracle inequalities. Preprint of the Strasbourg Louis Pasteur University, IRMA, 2007/09. Available at: http://hal.archives-ouvertes.fr/hal/-00179856/fr/ · Zbl 1154.62329
[8] Galtchouk, L.; Pergamenshchikov, S., Sharp non-asymptotic oracle inequalities for nonparametric heteroscedastic regression models, Journal of nonparametric statistics, 21, 1, 1-18, (2009) · Zbl 1154.62329
[9] Galtchouk, L., & Pergamenshchikov, S. (2008). Adaptive asymptotically efficient estimation in heteroscedastic nonparametric regression via model selection. Available at: http://hal.archives-ouvertes.fr/hal-00326910/fr/ · Zbl 1293.62091
[10] Gill, R.D.; Levit, B.Y., Application of the Van trees inequality: A Bayesian cramér – rao bound, Bernoulli, 1, 59-79, (1995) · Zbl 0830.62035
[11] Goldfeld, S.M.; Quandt, R.E., Nonlinear methods in econometrics, (1972), North-Holland London · Zbl 0231.62114
[12] Golubev, G.K.; Nussbaum, M., Adaptive spline estimates in a nonparametric regression model, Theory of probabability and its applications, 37, 521-529, (1993) · Zbl 0787.62044
[13] Ibragimov, I.A.; Hasminskii, R.Z., Statistical estimation: asymptotic theory, (1981), Springer New York
[14] Katz, D.; D’Argenio, D., Experimental design for estimating integrals by numerical quadrature, with applications to pharmacokinetic studies, Biometrics, 39, 621-628, (1983) · Zbl 0522.62094
[15] Kolmogorov, A.N.; Fomin, S.V., Elements of the theory of functions and functional analysis, (1989), Nauka Moscow · Zbl 0672.46001
[16] Nussbaum, M., Spline smoothing in regression models and asymptotic efficiency in \(\mathbf{L}_2\), Annals of statistics, 13, 984-997, (1985) · Zbl 0596.62052
[17] Pinsker, M.S., Optimal filtration of square integrable signals in Gaussian white noise, Problems of transmission information, 17, 120-133, (1981) · Zbl 0452.94003
[18] Stone, C.J., Optimal global rates of convergence for nonparametric regression, Annals of statistics, 10, 1040-1053, (1982) · Zbl 0511.62048
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.