Prediction model-based kernel density estimation when group membership is subject to missing

The density function is a fundamental concept in data analysis. When a population consists of heterogeneous subjects, it is often of great interest to estimate the density functions of the subpopulations. Nonparametric methods such as kernel smoothing estimates may be applied to each subpopulation to estimate the density functions if there are no missing values. In situations where the membership for a subpopulation is missing, kernel smoothing estimates using only subjects with membership available are valid only under missing complete at random (MCAR). In this paper, we propose new kernel smoothing methods for density function estimates by applying prediction models of the membership under the missing at random (MAR) assumption. The asymptotic properties of the new estimates are developed, and simulation studies and a real study in mental health are used to illustrate the performance of the new estimates.