  We introduce a statistical method for modeling and forecasting functional
panel data, where each element is a density. Density functions are nonnegative
and have a constrained integral and thus do not constitute a linear vector
space. We implement a center log-ratio transformation to transform densities
into unconstrained functions. These functions exhibit cross-sectionally
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-range covariance, is implemented.
By combining the forecasts of the time-varying residual component with the
deterministic trend component, we obtain h-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.
