Robust nonparametric regression estimation

In this paper we define a robust conditional location functional without requiring any moment condition. We apply the nonparametric proposals considered by C. Stone (Ann. Statist. 5 (1977), 595-645) to this functional equation in order to obtain strongly consistent, robust nonparametric estimates of the regression function. We give some examples by using nearest neighbor weights or weights based on kernel methods under no assumptions whatsoever on the probability measure of the vector (X,Y). We also derive strong convergence rates and the asymptotic distribution of the proposed estimates.

[1]  L. Devroye,et al.  On the L1 convergence of kernel estimators of regression functions with applications in discrimination , 1980 .

[2]  M. C. Jones,et al.  Spline Smoothing and Nonparametric Regression. , 1989 .

[3]  P. J. Huber.  Robust Estimation of a Location Parameter , 1964 .

[4]  R. Gray,et al.  Robustness of Estimators on Stationary Observations , 1979 .

[5]  L. Devroye,et al.  Distribution-Free Consistency Results in Nonparametric Discrimination and Regression Function Estimation , 1980 .

[6]  Y. Mack,et al.  Local Properties of k-NN Regression Estimates , 1981 .

[7]  L. Devroye.  On the Almost Everywhere Convergence of Nonparametric Regression Function Estimates , 1981 .

[8]  G. S. Watson,et al.  Smooth regression analysis , 1964 .

[9]  G. Collomb.  Estimation Non-paramétrique de la Régression: Revue Bibliographique@@@Estimation Non-parametrique de la Regression: Revue Bibliographique , 1981 .

[10]  E. Nadaraya.  On Estimating Regression , 1964 .

[11]  G. Collomb.  Estimation de la regression par la methode des k points les plus proches avec noyau : quelques propriétés de convergence ponctuelle , 1980 .

[12]  R. Maronna.  Robust $M$-Estimators of Multivariate Location and Scatter , 1976 .

[13]  L. Devroye.  Necessary and sufficient conditions for the pointwise convergence of nearest neighbor regression function estimates , 1982 .

[14]  W. Härdle.  Robust regression function estimation , 1984 .

[15]  R. Beran.  Robust Location Estimates , 1977 .

[16]  W. Stute.  Asymptotic Normality of Nearest Neighbor Regression Function Estimates , 1984 .

[17]  C. Spiegelman,et al.  Consistent Window Estimation in Nonparametric Regression , 1980 .

[18]  C. J. Stone,et al.  Consistent Nonparametric Regression , 1977 .

[19]  A. Krzyżak,et al.  Distribution-Free Pointwise Consistency of Kernel Regression Estimate , 1984 .

[20]  Shie-Shien Yang,et al.  Linear Functions of Concomitants of Order Statistics with Application to Nonparametric Estimation of a Regression Function , 1981 .

[21]  Ricardo Fraiman,et al.  Qualitative Robustness for Stochastic Processes , 1987 .

[22]  S. Yakowitz,et al.  Contributions to the Theory of Nonparametric Regression, with Application to System Identification , 1979 .

[23]  J. Staniswalis.  The Kernel Estimate of a Regression Function in Likelihood-Based Models , 1989 .