Density estimation and comparison are the fundamental problems in statistical inference. When parametric distribution assumptions are further made, the problems become estimation and comparison of the parameters of interest. In the situation that the parameters of interest are indexed by a set of covariates, regression is the most powerful tool for estimation and inference. In many situations, the densities are multimodal and parametric assumptions are difficult to make. In these cases, nonparametric density estimation methods are preferred. In this paper, we propose a functional linear model for the situation where a group of densities are indexed by a set of covariates. The basic unit of the data analysis is a density. Through a logistic density transformation, the relationship between the densities and covariates are modeled by a varying coefficient model. This is an extension of regression to a nonparametric setting in which no parametric distribution assumption is needed. Similar to the regression setting, we can borrow information across units in the estimation, make inference on the covariate effects, and make prediction on a new unit. We term the proposed model "density regression model" (DENREG). The method is illustrated by a numerical example and a nerve fiber density data.
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