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Improved model selection method for a regression function with dependent noise. (English) Zbl 1122.62023

Summary: This paper is devoted to nonparametric estimation, through the \(\mathcal{L}_2\)-risk, of a regression function based on observations with spherically symmetric errors, which are dependent random variables (except in the normal case). We apply a model selection approach using improved estimates. In a nonasymptotic setting, an upper bound for the risk is obtained (oracle inequality). Moreover asymptotic properties are given, such as upper and lower bounds for the risk, which provide optimal rates of convergence for penalized estimators.

MSC:

62G08 Nonparametric regression and quantile regression
62H12 Estimation in multivariate analysis
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[1] Barron A., Birgé L., Massart P. (1999). Risk bounds for model selection via penalization. Probability Theory and Related Fields, 113: 301–413 · Zbl 0946.62036
[2] Beran J. (1992). Statistical methods for data with long-range depedence. Statistical Science, 7: 404–427
[3] Berger J.O. (1975). Minimax estimation of location vectors for a wide class of densities. The Annals of Statistics, 3(6): 1318–1328 · Zbl 0322.62009
[4] Blanchard D., Fourdrinier D. (1999). Non trivial solutions of non-linear partial differential inequations and order cut-off. Rendiconti di Matematica, Serie VII, 19: 137–154 · Zbl 0947.35180
[5] Brown L.D. (1966). On the admissibility of invariant estimators of one or more location parameters. The Annals of Mathematical Statistics, 37: 1087–1136 · Zbl 0156.39401
[6] Csörgó S., Mielniczuk J. (1995). Nonparametric regression under longe range dependent normal errors. The Annals of Statistics, 23(3): 1000–1014 · Zbl 0852.62035
[7] Dahlhaus R. (1995). Efficient location and regression estimation for longrange dependent regression models. The Annals of Statistics, 23(3): 1029–1047 · Zbl 0838.62084
[8] Donoho D.L., Johnstone I.M., Kerkyacharian G., Picard D. (1995). Wavelet shrinkage: asymptotia?. Journal of Royal Statistical Society, Services B, 57: 301–369 · Zbl 0827.62035
[9] Efroimovich S.Yu. (1999). Nonparametric curve estimation. Methods, theory and applications. Springer, Berlin Haidelberg New York
[10] Fang K.-T., Kotz S., Ng K.-W. (1989). Symmetric Multivariate and Related Distributions. Chapman and Hall, New york
[11] Fourdrinier D., Strawderman W.E. (1996). A paradox concerning shrinkage estimators: should a known scale parameter be replaced by an estimated value in the shrinkage factor?. Journal of Multivariate Analysis, 59(2): 109–140 · Zbl 0864.62037
[12] Fourdrinier D., Wells M.T. (1994). Comparaison de procédures de sélection d’un modèle de régression: une approche décisionnelle. Comptes Rendus de l’Académie des Sciences Paris, v. 319, série I, p. 865–870
[13] Gunst R.F., Mason R.L. (1980). Regression analysis and its applications: a data oriented approach. New York, Marcel Dekker · Zbl 0433.62041
[14] James W., Stein C. (1961). Estimation with quadratic loss. In Proceedings of the Fourth Berkeley Symposium Mathematics, Statistics and Probability, University of California Press, Berkeley, 1: 361–380 · Zbl 1281.62026
[15] Hall P., Hart J. (1990). Nonparametric regression with long range dependence. Stochastics Processes and their Applications 36: 339–351 · Zbl 0713.62048
[16] Kariya T., Sinha B.K. (1993). The robustness of statistical tests. Academic, New York · Zbl 0699.62033
[17] Kotz S. (1975). Multivariate distributions at a cross-road. In Patil G.J., Kotz S., Ord J.K. Statistical Distributions in Scientific Work. (Eds.) 1: 245–270
[18] Nemirovski A. (2000). Topics in non-parametric statistics. Lectures on probability theory and statistics, Saint-Flour 1998. In Lecture Notes in Mathematics, 1738. Berlin Heidelberg New York: Springer, pp 85–277
[19] Stein C.M. (1956). Inadmissibility of usual estimator of the mean of a multivariate distribution. In Proceedings of the 3rd Berkely Symposium in Mathematical Statistics and Probability, 1 197–206 · Zbl 0073.35602
[20] Stein C.M. (1981). Estimation of the mean of a multivariate normal distribution. The Annals of Statistics, 9: 1135–1151 · Zbl 0476.62035
[21] Timan A.F. (1963). Theory of approximation of functions of a real variable. Pergamon, Oxford · Zbl 0117.29001
[22] Tsybakov A. (1998). Pointwise and sup-norm sharp adaptive estimation of functions on the Sobolev classes. The Annals of Statistics, 26(6): 221–245 · Zbl 0933.62028
[23] Ziemer W.P. (1989). Weakly differentiable functions–sobolev spaces and functions of bounded variation. Springer, Berlin Heidelberg New York · Zbl 0692.46022
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