Bayesian Semiparametric Density Deconvolution in the Presence of Conditionally Heteroscedastic Measurement Errors

• Published in: Journal of computational and graphical statistics Vol. 23; no. 4; pp. 1101 - 1125
• Main Authors: Sarkar, Abhra, Mallick, Bani K., Staudenmayer, John, Pati, Debdeep, Carroll, Raymond J.
• Format: Journal Article
• Language: English
• Published: United States Taylor & Francis 02.10.2014; American Statistical Association, Institute of Mathematical Statistics, and Interface Foundation of North America; Taylor & Francis Ltd
• Subjects: B-spline; Bayesian analysis; Bayesian Methods; Conditional heteroscedasticity; Density; Density estimation; Dirichlet problem; Dirichlet process mixture models; Epidemiology; Error rates; Estimating techniques; Estimation methods; Mathematical models; Measurement errors; Nonparametric models; Parametric models; Random variables; Sample size; Simulations; Skew-normal distribution; Statistical variance; Studies; Variance function; Measurement errors; Variance function; Dirichlet process mixture models; Skew-normal distribution; B-spline; Density deconvolution; Conditional heteroscedasticity
• ISSN: 1061-8600; 1537-2715
• DOI: 10.1080/10618600.2014.899237

We consider the problem of estimating the density of a random variable when precise measurements on the variable are not available, but replicated proxies contaminated with measurement error are available for sufficiently many subjects. Under the assumption of additive measurement errors this reduces to a problem of deconvolution of densities. Deconvolution methods often make restrictive and unrealistic assumptions about the density of interest and the distribution of measurement errors, for example, normality and homoscedasticity and thus independence from the variable of interest. This article relaxes these assumptions and introduces novel Bayesian semiparametric methodology based on Dirichlet process mixture models for robust deconvolution of densities in the presence of conditionally heteroscedastic measurement errors. In particular, the models can adapt to asymmetry, heavy tails, and multimodality. In simulation experiments, we show that our methods vastly outperform a recent Bayesian approach based on estimating the densities via mixtures of splines. We apply our methods to data from nutritional epidemiology. Even in the special case when the measurement errors are homoscedastic, our methodology is novel and dominates other methods that have been proposed previously. Additional simulation results, instructions on getting access to the dataset and R programs implementing our methods are included as part of online supplementary materials.