Statistica Sinica 21 (2011), 1023-1063
doi:10.5705/ss.2009.039

ESTIMATION OF OBSERVATION-ERROR VARIANCE

IN ERRORS-IN-VARIABLES REGRESSION

Aurore Delaigle$^{1}$ and Peter Hall$^{1,2}$

$^1$The University of Melbourne and $^2$University of California at Davis

Abstract: Assessing the variability of an estimator is a key component of the process of statistical inference. In nonparametric regression, estimating observation-error variance is the principal ingredient needed to estimate the variance of the regression mean. Although there is an extensive literature on variance estimation in nonparametric regression, the techniques developed in conventional settings generally cannot be applied to the problem of regression with errors in variables, where the explanatory variables are not directly observable. In this paper we introduce methods for estimating observation-error variance in errors-in-variables regression. We consider cases where the variance is modelled either nonparametrically or parametrically. The performance of our methods is assessed both numerically and theoretically. We also suggest a fully data-driven bandwidth selection procedure, a problem that is notoriously difficult in errors-in-variables contexts.

Key words and phrases: Bandwidth, kernel estimation, nonparametric curve estimation, nonparametric regression, parametric model, statistical smoothing, variance estimation.