Density estimation in the presence of heteroscedastic measurement error

We consider density estimation when the variable of interest is subject to heteroscedastic measurement error. The density is assumed to have a smooth but unknown functional form that we model with a penalized mixture of B-splines. We treat the situation in which multiple mismeasured observations of each variable 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 article's main contributions are to address the effects of heteroscedastic measurement error effectively, explain the biases caused by ignoring heteroscedasticity, and present an equivalent kernel for a spline-based density estimator. Derivation of the equivalent kernel may be of independent interest. We use small-sigma asymptotics to approximate the biases incurred by assuming that the measurement error is homoscedastic when it actually is heteroscedastic. The biases incurred by misspecifying heteroscedastic measurement error as homoscedastic can be substantial. We fit the model using Bayesian methods and apply it to an example from nutritional epidemiology and a simulation experiment.