Abstract

Statistical models for samples of (one-dimensional) distribution-valued stochastic processes are of interest for longitudinally observed distributions such as age-at-death distributions in demography and are part of the emerging area of Distributional Data Analysis. While functional data analysis provides a toolbox for the analysis of samples of processes that take values in Euclidean spaces, there is at present no coherent statistical methodology available for samples of distribution-valued processes.
To address this challenge, we introduce a transport model for samples of distribution-valued stochastic processes that implements an intrinsic approach,
whereby the observed distributions are converted to optimal transports from the respective barycenter. Substituting transports for distributions addresses the challenge of centering distribution-valued processes and leads to
a useful and interpretable representation of each realized process by an overall time-independent transport and a real-valued trajectory. This representation is obtained by utilizing a recently introduced scalar multiplication for transports {Zhu, M. JRSSB 2023) and facilitates a connection to Gaussian processes. This connection makes it possible to include longitudinal scenarios where the distribution-valued processes are only observed on a sparse grid of randomly situated time points. The proposed approach is supported by convergence results and its practical utility in applications. This talk is based on joint work with Hang Zhou.