Estimation of Distribution Functions in Measurement Error Models

Many practical problems are related to the pointwise estimation of dis- tribution functions when data contains measurement errors. Motivation for these problems comes from diverse fields such as astronomy, reliability, quality control, public health and survey data. Recently, Dattner, Goldenshluger and Juditsky (2011) showed that an estimator based on a direct inversion formula for distribution functions has nice properties when the tail of the characteristic function of the mea- surement error distribution decays polynomially. In this paper we derive theoretical properties for this estimator for the case where the error distri- bution is smoother and study its finite sample behavior for different error distributions. Our method is data-driven in the sense that we use only known information, namely, the error distribution and the data. Applica- tion of the estimator to estimating hypertension prevalence based on real data is also examined.