×

Accelerated convergence for nonparametric regression with coarsened predictors. (English) Zbl 1129.62032

Summary: We consider nonparametric estimation of a regression function for a situation where precisely measured predictors are used to estimate the regression curve for coarsened, that is, less precise or contaminated predictors. Specifically, while one has available a sample \((W_{1}, Y_{1}), \dots , (W_n, Y_n)\) of independent and identically distributed data, representing observations with precisely measured predictors, where \(E(Y_i|W_i)=g(W_i)\), instead of the smooth regression function \(g\), the target of interest is another smooth regression function \(m\) that pertains to predictors \(X_i\) that are noisy versions of the \(W_i\). Our target is then the regression function \(m(x)=E(Y|X=x)\), where \(X\) is a contaminated version of \(W\), that is, \(X=W+\delta \). It is assumed that either the density of the errors is known, or replicated data are available resembling, but not necessarily the same as, the variables \(X\).
In either case, and under suitable conditions, we obtain \(\sqrt n\)-rates of convergence of the proposed estimator and its derivatives, and establish a functional limit theorem. Weak convergence to a Gaussian limit process implies pointwise and uniform confidence intervals and \(\sqrt n\)-consistent estimators of extrema and zeros of \(m\). It is shown that these results are preserved under more general models in which \(X\) is determined by an explanatory variable. Finite sample performance is investigated in simulations and illustrated by a real data example.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
60F17 Functional limit theorems; invariance principles

Software:

ElemStatLearn
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Bickel, P. J., Klaassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD. · Zbl 0786.62001
[2] Carroll, R. J. and Hall, P. (2004). Low-order approximations in deconvolution and regression with errors in variables. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 31-46. · Zbl 1062.62066
[3] Carroll, R. J., Maca, J. D. and Ruppert, D. (1999). Nonparametric regression in the presence of measurement error. Biometrika 86 541-554. · Zbl 0938.62039
[4] Carroll, R. J., Ruppert, D. and Stefanski, L. (1995). Measurement Error in Nonlinear Models. Chapman and Hall, London. · Zbl 0853.62048
[5] Devanarayan, V. and Stefanski, L. A. (2002). Empirical simulation extrapolation for measurement error models with replicate measurements. Statist. Probab. Lett. 59 219-225. · Zbl 1092.62558
[6] Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems. Ann. Statist. 19 1257-1272. · Zbl 0729.62033
[7] Fan, J. and Masry, E. (1992). Multivariate regression estimation with errors-in-variables: Asymptotic normality for mixing processes. J. Multivariate Anal. 43 237-271. · Zbl 0769.62028
[8] Fan, J. and Truong, Y. K. (1993). Nonparametric regression with errors in variables. Ann. Statist. 21 1900-1925. · Zbl 0791.62042
[9] Fan, J., Truong, Y. K. and Wang, Y. (1991). Nonparametric function estimation involving errors-in-variables. In Nonparametric Functional Estimation and Related Topics (G. Roussas, ed.) 613-627. Kluwer, Dordrecht.
[10] Hastie, T., Tibshirani, R. and Friedman, J. (2001). The Elements of Statistical Learning. Data Mining, Inference and Prediction. Springer, New York. · Zbl 0973.62007
[11] Hobert, J. P. and Wand, M. P. (2000). Automatic generalized nonparametric regression via maximum likelihood. Technical report, Dept. Biostatistics, Harvard School of Public Health.
[12] Ioannides, D. A. and Matzner-Løber, E. (2002). Nonparametric estimation of the conditional mode with errors-in-variables: Strong consistency for mixing processes. J. Nonparametr. Statist. 14 341-352. · Zbl 1014.62102
[13] Li, T. and Vuong, Q. (1998). Nonparametric estimation of the measurement error model using multiple indicators. J. Multivariate Anal. 65 139-165. · Zbl 1127.62323
[14] Linton, O. and Whang, Y.-J. (2002). Nonparametric estimation with aggregated data. Econometric Theory 18 420-468. · Zbl 1109.62312
[15] Stefanski, L. and Carroll, R. J. (1990). Deconvoluting kernel density estimators. Statistics 21 169-184. · Zbl 0697.62035
[16] Stefanski, L. A. and Cook, J. R. (1995). Simulation-extrapolation: The measurement error jackknife. J. Amer. Statist. Assoc. 90 1247-1256. · Zbl 0868.62062
[17] Taupin, M. L. (2001). Semi-parametric estimation in the nonlinear structural errors-in-variables model. Ann. Statist. 29 66-93. · Zbl 1029.62039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.