On the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document} norms of kernel regression estimators for

We consider kernel methods to construct nonparametric estimators of a regression function based on incomplete data. To tackle the presence of incomplete covariates, we employ Horvitz–Thompson-type inverse weighting techniques, where the weights are the selection probabilities. The unknown selection probabilities are themselves estimated using (1) kernel regression, when the functional form of these probabilities are completely unknown, and (2) the least-squares method, when the selection probabilities belong to a known class of candidate functions. To assess the overall performance of the proposed estimators, we establish exponential upper bounds on the $$L_p$$Lp norms, $$1\le p<\infty $$1≤p<∞, of our estimators; these bounds immediately yield various strong convergence results. We also apply our results to deal with the important problem of statistical classification with partially observed covariates.

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

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

[3]  Majid Mojirsheibani.  Nonparametric curve estimation with missing data: A general empirical process approach , 2007 .

[4]  G. Imbens,et al.  Efficient Estimation of Average Treatment Effects Using the Estimated Propensity Score , 2000 .

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

[6]  D. Rubin,et al.  Statistical Analysis with Missing Data. , 1989 .

[7]  H. Y. Chen.  Nonparametric and Semiparametric Models for Missing Covariates in Parametric Regression , 2004 .

[8]  Adam Krzyzak,et al.  A Distribution-Free Theory of Nonparametric Regression , 2002, Springer series in statistics.

[9]  Harro Walk.  Almost Sure Convergence Properties of Nadaraya-Watson Regression Estimates , 2005 .

[10]  Joseph G. Ibrahim,et al.  Missing covariates in generalized linear models when the missing data mechanism is non‐ignorable , 1999 .

[11]  M. Wand,et al.  Variational Bayesian Inference for Parametric and Nonparametric Regression With Missing Data , 2011 .

[12]  Qianqian Zhu,et al.  AN EFFECTIVE TECHNIQUE OF MULTIPLE IMPUTATION IN NONPARAMETRIC QUANTILE REGRESSION , 2014 .

[13]  A E Bostwick,et al.  THE THEORY OF PROBABILITIES. , 1896, Science.

[14]  Roderick J. A. Little,et al.  Statistical Analysis with Missing Data: Little/Statistical Analysis with Missing Data , 2002 .

[15]  Joseph G. Ibrahim,et al.  A conditional model for incomplete covariates in parametric regression models , 1996 .

[16]  D. Horvitz,et al.  A Generalization of Sampling Without Replacement from a Finite Universe , 1952 .

[17]  H. Rockette,et al.  On maximum likelihood estimation in parametric regression with missing covariates , 2005 .

[18]  M. Mojirsheibani.  Some results on classifier selection with missing covariates , 2010, Metrika.

[19]  Jeffrey S. Racine,et al.  Nonparametric estimation of regression functions with both categorical and continuous data , 2004 .

[20]  D. Pollard.  Convergence of stochastic processes , 1984 .

[21]  Samiran Sinha,et al.  Semiparametric approach for non‐monotone missing covariates in a parametric regression model , 2014, Biometrics.

[22]  Harro Walk.  On cross-validation in kernel and partitioning regression estimation , 2002 .

[23]  L. Devroye,et al.  An equivalence theorem for L1 convergence of the kernel regression estimate , 1989 .

[24]  L. Devroye,et al.  Nonparametric density estimation : the L[1] view , 1987 .

[25]  Andrea Rotnitzky,et al.  Nonparametric Regression With Missing Outcomes Using Weighted Kernel Estimating Equations , 2010, Journal of the American Statistical Association.

[26]  Sam Efromovich,et al.  Nonparametric Regression With Predictors Missing at Random , 2011 .

[27]  W. Härdle,et al.  Optimal Bandwidth Selection in Nonparametric Regression Function Estimation , 1985 .

[28]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[29]  S. Geer,et al.  High-dimensional additive modeling , 2008, 0806.4115.

[30]  Jeffrey S. Racine,et al.  Nonparametric Econometrics: The np Package , 2008 .

[31]  J. Robins,et al.  Estimation of Regression Coefficients When Some Regressors are not Always Observed , 1994 .

[32]  Lixing Zhu,et al.  Multi-index regression models with missing covariates at random , 2014, J. Multivar. Anal..

[33]  Francesco Bravo,et al.  Semiparametric estimation with missing covariates , 2015, J. Multivar. Anal..

[34]  A. Krzyżak,et al.  Strong consistency of automatic kernel regression estimates , 2003 .

[35]  James M. Robins,et al.  Estimation in Partially Linear Models With Missing Covariates , 2004 .