Regression-Assisted Deconvolution

We present a semi-parametric deconvolution estimator for the density function of a random variable X that is measured with error. Traditional deconvolution estimators rely only on assumptions about the distribution of X and the error in its measurement, and ignore information available in auxiliary variables. Our method assumes the availability of a covariate vector statistically related to X by a mean- variance function regression model, where regression errors are normally distributed and independent of the measurement errors. Under common parametric assumptions on the conditional mean and variance, the estimator is √n-consistent for the true density of X, a substantial improvement over the logarithmic rates achieved by non- parametric deconvolution estimators. Simulations suggest that the estimator achieves a much lower integrated squared error than the observed-data kernel density estimator when models are correctly specified and the assumption of normal regression errors is met. We illustrate the method using anthropometric measurements of newborns to estimate the density function of newborn length.