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, and the measurement error is assumed to be additive and normal. The paper’s main contributions to this problem are to address the effects of heteroskedastic measurement error, to explain the biases caused by ignoring heteroskedasticity, and to present an approximate equivalent kernel for a spline based density estimator. The derivation of the equivalent kernel may be of independent interest. We model the variance as a smooth function of the unobserved variable of interest, and we use small-σ asymptotics to describe the biases incurred by assuming the measurement error is homoskedastic when it actually is heteroskedastic. We fit the model using Bayesian methods. An example from nutritional epidemiology and an example that uses simulated data are included.

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