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Asymptotically efficient estimators for nonparametric heteroscedastic regression models. (English) Zbl 1220.62039

Summary: This paper concerns the estimation of a function at a point in nonparametric heteroscedastic regression models with Gaussian noise or noise having unknown distribution. In those cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62G07 Density estimation
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References:

[1] Billingsley, P., Convergence of probability measures, wiley series in probability and statistics: probability and statistics, (1999), John Wiley & Sons Inc New York, A Wiley-Interscience Publication
[2] Donoho, D.L., Asymptotic minimax risk for sup-norm loss: solution via optimal recovery, Probab. theory related fields, 99, 2, 145-170, (1994) · Zbl 0802.62007
[3] Donoho, D.L.; Liu, R.C., Geometrizing rates of convergence. III, Ann. statist., 19, 668-701, (1991) · Zbl 0754.62029
[4] Efromovich, S., Sequential design and estimation in heteroscedastic nonparametric regression, Sequential anal., 26, 1, 3-25, (2007) · Zbl 1112.62080
[5] Efromovich, S.; Pinsker, M., Sharp-optimal and adaptive estimation for heteroscedastic nonparametric regression, Statist. sinica, 6, 4, 925-942, (1996) · Zbl 0857.62037
[6] Freedman, D., Brownian motion and diffusion, (1971), Holden Day San Francisco · Zbl 0231.60072
[7] L. Galtchouk, S. Pergamenshchikov, Efficient adaptive nonparametric estimation in heteroscedastic regression models, Preprint of the Strasbourg Louis Pasteur University, IRMA, 2005, available online at http://hal.archives-ouvertes.fr/hal-00129707/fr/ · Zbl 1293.62091
[8] Galtchouk, L.; Pergamenshchikov, S., Asymptotically efficient estimates for nonparametric regression models, Statist. probab. lett., 76, 852-860, (2006) · Zbl 1089.62044
[9] Goldfeld, S.; Quandt, R., Nonlinear methods in econometrics, (1972), North-Holland Amsterdam, London
[10] Ibragimov, I.A.; Has’minskii, R.Z., Statistical estimation: asymptotic theory, (1981), Springer Berlin, New York
[11] Korostelev, A., Exact asymptotically minimax estimator for nonparametric regression in uniform norm, Theory probab. appl., 38, 737-743, (1993)
[12] Sacks, J.; Ylvisaker, D., Asymptotically optimum kernels for density estimation at a point, Ann. statist., 9, 2, 334-346, (1981) · Zbl 0458.62031
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