Empirical likelihood in a regression model with noised variables

This paper is concerned with a recently developed regression model with noised variables in which the means of the response and some covariable components are nonparametric functions of an auxiliary variable. Previous results have shown that the de-noised estimators of the parameters of interest are asymptotically normal when undersmoothing is applied. But undersmoothing causes difficulties in bandwidth selection. To avoid this problem, we propose an empirical log-likelihood ratio for the regression coefficients and derive a nonparametric version of Wilk's theorem. The confidence region based on the empirical likelihood has three advantages compared with those based on asymptotic normality: (1) It does not have the predetermined symmetry, which enables it to better reflect the true shape of the underlying distribution; (2) it does not involve any asymptotic covariance matrix estimation and hence is robust against the heteroscedasticity; and (3) it avoids undersmoothing the regressor functions so that optimal bandwidth can be used. A small simulation is conducted to compare the finite sample performances of these two methods. An example of application on a set of advertising data is also illustrated.