Optimal model averaging for global Fréchetregression

Global Fréchet regression provides a powerful tool for studying complex non-Euclidean data, which are frequently encountered in many areas of applications. However, it is often difficult and challenging for global Fréchet regression to determine the covariates in practice. To alleviate the issue of model selection uncertainty, we propose a feasible model averaging approach coupled with a data-driven cross-validation weight choice criterion. Theoretically, we establish the consistency of the global Fréchet regression estimators and derive the optimal pointwise convergence rate and a near-optimal uniform convergence rate even when the candidate models are misspecified. Furthermore, we establish the asymptotic optimality under the model misspecification and the rate of convergence for the weight estimator obtained by cross-validation. In addition, if the candidate model set includes at least one correct model, the weights chosen by cross-validation are shown to be asymptotically concentrated on all correct models. Moreover, based on the proposed model averaging approach, we develop a reliable variable importance procedure to measure the importance of each covariate in global Fréchet regression. This procedure is proven to asymptotically identify all the variables in the true model. Simulation experiments and an empirical application illustrate the superiority of the proposed model averaging method over some common model selection and averaging methods.