Abstract
A new estimator of a regression function is introduced via minimizing the L 1-distance between some empirical function and its theoretical counterpart plus penalty for the roughness. The L 1-risk of the estimator is bounded from above for every sample size no matter what the dependence structure of the observed random variables is. In the case of independent errors of measurement with a common variance the estimator is shown to achieve the optimal L 1-rate of convergence within the class of m-times differentiable functions with bounded derivatives.
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References
Dodge, Y. (1987). Statistical Data Analysis Based on the L1-norm and Related Methods, North-Holland, Tokyo.
Gajek, L. (1989). Estimating a density and its derivatives via the minimum distance method, Probab. Theory Related Fields, 80, 601–617.
Gajek, L. (1990). The rate of convergence of the nonparametric minimum L 1-penalized distance estimator, Probab. Math. Statist. (to appear).
Jennen-Steinmetz, Ch. and Gasser, T. (1988). A unifying approach to nonparametric regression estimation, J. Amer. Statist. Assoc., 83, 1084–1089.
Pollard, D. (1990). Asymptotics for least absolute deviation regression estimators (preprint).
Silverman, D. (1984). Spline smoothing: the equivalent variable kernel method, Ann. Statist., 12, 898–916.
Silverman, D. (1985). Some aspects of the spline smoothing approach to nonparametric regression curve fitting (with discussion), Journal of the Royal Statistical Association, B, 47, 1–52.
van de Geer, S. (1990). Estimating a regression function, Ann. Statist., 18, 907–924.
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Gajek, L., Kaluszka, M. Upper bounds for the L 1-risk of the minimum L 1-distance regression estimator. Ann Inst Stat Math 44, 737–744 (1992). https://doi.org/10.1007/BF00053402
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DOI: https://doi.org/10.1007/BF00053402