Nonparametric Density Estimation of a Long-Term Trend from Repeated Semicontinuous Data

We consider nonparametric estimation of the density of the long-term trend of a semicontinuous variable observed repeatedly over time. These variables arise when measuring the intensity of an intermittent phenomenon, such as the intake of an episodically consumed nutrient or the concentration of an intermittent toxic substance: when the phenomenon is absent, the measurement is equal to zero; otherwise, it is positive. Semicontinuous data are usually represented by a two-part model describing the zeros and the nonzeros separately, often under parametric assumptions. Recently, Camirand Lemyre, Carroll, and Delaigle showed that it is possible to relax the distributional assumptions on the part that models the nonzeros, but like other existing work, they used a parametric model for the conditional probability $H$ of observing a nonzero value. We develop a nonparametric estimator of $H$ and of the density of the long-term trend. We illustrate our method on simulated examples and apply it to estimate the density of long-term fruit intake, using data from the Eating at America's Table Study. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.