Estimating smooth distribution function in the presence of heteroscedastic measurement errors.

Measurement error occurs in many biomedical fields. The challenges arise when errors are heteroscedastic since we literally have only one observation for each error distribution. This paper concerns the estimation of smooth distribution function when data are contaminated with heteroscedastic errors. We study two types of methods to recover the unknown distribution function: a Fourier-type deconvolution method and a simulation extrapolation (SIMEX) method. The asymptotics of the two estimators are explored and the asymptotic pointwise confidence bands of the SIMEX estimator are obtained. The finite sample performances of the two estimators are evaluated through a simulation study. Finally, we illustrate the methods with medical rehabilitation data from a neuro-muscular electrical stimulation experiment.