Session: Technical Session I

Date & Time: Friday, May 22 — 11:00 AM–12:30 PM

Talk Title

Metric Statistics: Basic Tools and Distribution-valued Processes

Abstract

Statistical methods for data situated in general metric spaces are not yet well developed. Such data have been referred to as Random Objects and the field dealing with them as Metric Statistics. Relevant basic tools include measures of location and spread as well as regression models. After reviewing these basics, we will discuss models for samples of distribution-valued stochastic processes that are of interest for longitudinally observed one-dimensional distributions such as age-at-death distributions in demography. Distributions can be viewed as elements of a metric space such as the Wasserstein space. While functional data analysis provides a toolbox for the analysis of samples of processes that take values in Euclidean spaces, with foundational contributions by Peter Hall, there is at present no coherent statistical methodology available for samples of distribution-valued processes, for which we introduce an intrinsic optimal transport model. In a first centering step, the observed distributions are converted to optimal transports from the respective barycenter. We then propose an interpretable representation of each realized process by decomposing it into a time-independent transport and a real-valued random trajectory, utilizing a recent transport algebra to establish a connection with real-valued Gaussian processes. This connection makes it possible to consider not only continuously sampled distributions but also sparse/longitudinal sampling designs. This talk is based on joint work with Hang Zhou (University of North Carolina).