  It is important to properly correct for measurement error when estimating
density functions associated with biomedical variables. These estimators that
adjust for measurement error are broadly referred to as density deconvolution
estimators. While most methods in the literature assume the distribution of the
measurement error to be fully known, a recently proposed method based on the
empirical phase function (EPF) can deal with the situation when the measurement
error distribution is unknown. The EPF density estimator has only been
considered in the context of additive and homoscedastic measurement error;
however, the measurement error of many biomedical variables is heteroscedastic
in nature. In this paper, we developed a phase function approach for density
deconvolution when the measurement error has unknown distribution and is
heteroscedastic. A weighted empirical phase function (WEPF) is proposed where
the weights are used to adjust for heteroscedasticity of measurement error. The
asymptotic properties of the WEPF estimator are evaluated. Simulation results
show that the weighting can result in large decreases in mean integrated
squared error (MISE) when estimating the phase function. The estimation of the
weights from replicate observations is also discussed. Finally, the
construction of a deconvolution density estimator using the WEPF is compared to
an existing deconvolution estimator that adjusts for heteroscedasticity, but
assumes the measurement error distribution to be fully known. The WEPF
estimator proves to be competitive, especially when considering that it relies
on the minimal assumption of the distribution of measurement error.
