Density estimation in the presence of heteroskedastic measurement error

We consider density estimation when the variable of interest is subject to heteroskedastic measurement error. The density is assumed to have a smooth but unknown functional form which we model with a penalized mixture of B-splines. We treat the situation where multiple mismeasured observations of each of the variables of interest are observed for at least some of the subjects, and the measurement error is assumed to be additive and normal. The measurement error variance function is modeled with a second penalized mixture of B-splines. The paper’s main contributions are to address the effects of heteroskedastic measurement error effectively, to explain the biases caused by ignoring heteroskedasticity, and to present an equivalent kernel for a spline based density estimator. The derivation of the equivalent kernel may be of independent interest. We use small-sigma asymptotics to approximate the biases incurred by assuming the measurement error is homoskedastic when it actually is heteroskedastic. The biases incurred by misspecifying heteroskedastic measurement error as homoskedastic can be substantial. We fit the model using Bayesian methods. An example from nutritional epidemiology and a simulation experiment are included.