Title: Uniform confidence bands for nonparametric errors-in-variables regression,
Abstract: This paper develops a method to construct uniform confidence bands for a
nonparametric regression function where a predictor variable is subject to a
measurement error. We allow for the distribution of the measurement error to be
unknown, but assume that there is an independent sample from the measurement
error distribution. The sample from the measurement error distribution need not
be independent from the sample on response and predictor variables. The
availability of a sample from the measurement error distribution is satisfied
if, for example, either 1) validation data or 2) repeated measurements (panel
data) on the latent predictor variable with measurement errors, one of which is
symmetrically distributed, are available. The proposed confidence band builds
on the deconvolution kernel estimation and a novel application of the
multiplier (or wild) bootstrap method. We establish asymptotic validity of the
proposed confidence band under ordinary smooth measurement error densities,
showing that the proposed confidence band contains the true regression function
with probability approaching the nominal coverage probability. To the best of
our knowledge, this is the first paper to derive asymptotically valid uniform
confidence bands for nonparametric errors-in-variables regression. We also
propose a novel data-driven method to choose a bandwidth, and conduct
simulation studies to verify the finite sample performance of the proposed
confidence band. Applying our method to a combination of two empirical data
sets, we draw confidence bands for nonparametric regressions of medical costs
on the body mass index (BMI), accounting for measurement errors in BMI.
Finally, we discuss extensions of our results to specification testing, cases
with additional error-free regressors, and confidence bands for conditional
distribution functions.