Frequentist Model Averaging for Global Fréchet Regression

To consider model uncertainty in global Fréchet regression and improve density response prediction, we propose a frequentist model averaging method. The weights are chosen by minimizing a cross-validation criterion based on Wasserstein distance. In the cases where all candidate models are misspecified, we prove that the corresponding model averaging estimator has asymptotic optimality, achieving the lowest possible Wasserstein distance. When there are correctly specified candidate models, we prove that our method asymptotically assigns all weights to the correctly specified models. Numerical results of extensive simulations and a real data analysis on intracerebral hemorrhage data strongly favour our method.

I. Introduction

Data consisting of samples of probability density functions are increasingly prevalent in various scientific fields, such as biology, econometrics, and medical science. Examples include population age and mortality distributions across different countries or regions [1], [2], as well as the distributions of functional magnetic resonance imaging (MRI) scans in the brain [3]. Despite the growing popularity of probability density function data, statistical methods for analyzing such data are limited, with only a few existing works available [4], [5], [7], [8], [10]. The majority of current research focuses on methods for depicting the association between densities and Euclidean or non-Euclidean predictors through estimated conditional mean densities, which are defined as conditional Fréchet means under a suitable metric. However, similar to the traditional regression framework, much of the practical interest in Fréchet regression applications lies in prediction, rather than solely in the inherent density-predictor relationships.