Varying-Coefficient Additive Models with Density Responses and Functional Auto-Regressive Error Process

In many practical applications, data collected over time exhibit auto-correlation, which, if not addressed, can lead to incorrect statistical inferences. To address this, we propose a varying-coefficient additive model with density responses, incorporating a functional autoregressive (FAR) error process to account for serial dependency. We present a three-step spline-based estimation procedure for the varying-coefficient components after mapping densities into a linear space using the log-quantile density transformation. First, a B-spline series approximation provides initial estimates of the bivariate varying-coefficient functions. Second, spline estimation of the error process is obtained from the residuals. Lastly, improved estimates of the additive components are obtained by removing the estimated error process. Theoretical results, including convergence rates and asymptotic properties, are provided, and the practical performance of the method is demonstrated through simulations and real data analyses.