Forecasting Density‐Valued Functional Panel Data

Abstract

We introduce a statistical method for modelling and forecasting functional panel data represented by multiple densities. Density functions are non‐negative and have a constrained integral, and thus do not constitute a linear vector space. We implement a centre log‐ratio transformation to transform densities into unconstrained functions. These functions exhibit cross‐sectional correlation and temporal dependence. Via a functional analysis‐of‐variance decomposition, we decompose the unconstrained functional panel data into a deterministic trend component and a time‐varying residual component. To produce forecasts for the time‐varying component, a functional time series forecasting method, based on the estimation of the long‐run covariance, is implemented. By combining the forecasts of the time‐varying residual component with the deterministic trend component, we obtain ‐step‐ahead forecast curves for multiple populations. Illustrated by age‐ and sex‐specific life‐table death counts in the United States, we apply our proposed method to generate forecasts of the life‐table death counts for 51 states.