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It is important to properly correct for measurement error when estimatingdensity functions associated with biomedical variables. These estimators thatadjust for measurement error are broadly referred to as density deconvolutionestimators. While most methods in the literature assume the distribution of themeasurement error to be fully known, a recently proposed method based on theempirical phase function (EPF) can deal with the situation when the measurementerror distribution is unknown. The EPF density estimator has only beenconsidered in the context of additive and homoscedastic measurement error;however, the measurement error of many biomedical variables is heteroscedasticin nature. In this paper, we developed a phase function approach for densitydeconvolution when the measurement error has unknown distribution and isheteroscedastic. A weighted empirical phase function (WEPF) is proposed wherethe weights are used to adjust for heteroscedasticity of measurement error. Theasymptotic properties of the WEPF estimator are evaluated. Simulation resultsshow that the weighting can result in large decreases in mean integratedsquared error (MISE) when estimating the phase function. The estimation of theweights from replicate observations is also discussed. Finally, theconstruction of a deconvolution density estimator using the WEPF is compared toan existing deconvolution estimator that adjusts for heteroscedasticity, butassumes the measurement error distribution to be fully known. The WEPFestimator proves to be competitive, especially when considering that it relieson the minimal assumption of the distribution of measurement error.

Phase Function Density Deconvolution with Heteroscedastic Measurement Error of Unknown Type