Monte Carlo methods for nonparametric regression with heteroscedastic measurement error.

Nonparametric regression is a fundamental problem in statistics but challenging when the independent variable is measured with error. Among the first approaches was an extension of deconvoluting kernel density estimators for homescedastic measurement error. The main contribution of this article is to propose a new simulation-based nonparametric regression estimator for the heteroscedastic measurement error case. Similar to some earlier proposals, our estimator is built on principles underlying deconvoluting kernel density estimators. However, the proposed estimation procedure uses Monte Carlo methods for estimating nonlinear functions of a normal mean, which is different than any previous estimator. We show that the estimator has desirable operating characteristics in both large and small samples and apply the method to a study of benzene exposure in Chinese factory workers.

Regression-assisted deconvolution.

We present a semi-parametric deconvolution estimator for the density function of a random variable biX that is measured with error, a common challenge in many epidemiological studies. 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. 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.

Density Estimation with Replicate Heteroscedastic Measurements.

We present a deconvolution estimator for the density function of a random variable from a set of independent replicate measurements. We assume that measurements are made with normally distributed errors having unknown and possibly heterogeneous variances. The estimator generalizes the deconvoluting kernel density estimator of Stefanski and Carroll (1990), with error variances estimated from the replicate observations. We derive expressions for the integrated mean squared error and examine its rate of convergence as n → ∞ and the number of replicates is fixed. We investigate the finite-sample performance of the estimator through a simulation study and an application to real data.