Density-valued time series: Nonparametric density-on-density regression

Abstract

This paper is concerned with forecasting probability density functions. Density functions are nonnegative and have a constrained integral; thus, they do not constitute a vector space. Implementing unconstrained functional time-series forecasting methods is problematic for such nonlinear and constrained data. A novel forecasting method is developed based on a nonparametric function-on-function regression, where both the response and the predictor are probability density functions. Asymptotic properties of our nonparametric regression estimator are established, as well as its finite-sample performance through a series of Monte-Carlo simulation studies. Using COVID-19 data from the French department and age-specific period life tables from the United States, we assess and compare the finite-sample forecast accuracy of the proposed method with several existing methods.