Density‐Valued ARMA Models by Spline Mixtures

ABSTRACT This paper proposes a novel framework for modeling time series of probability density functions by extending autoregressive moving average (ARMA) models to density‐valued data. The method is based on a transformation approach, wherein each density function on a compact domain is approximated by a B‐spline mixture representation. Through generalized logit and softmax mappings, the space of density functions is transformed into an unconstrained Euclidean space, enabling the application of classical time series techniques. We define ARMA‐type dynamics in the transformed space. Estimation is carried out via least squares for density‐valued AR models and Whittle likelihood for ARMA models, with asymptotic normality derived under the joint divergence of the time horizon and basis dimension. The proposed methodology is applied to spatiotemporal human population data in Tokyo, where meaningful temporal structures in the distributional dynamics are successfully captured.