Doubly robust estimation with functional outcomes missing at random

TL;DR

This paper develops semi-parametric estimators for the mean of functional outcomes with missing data, establishing their asymptotic properties and demonstrating their effectiveness through simulations and an application.

Contribution

It introduces double robust estimators for functional data with missing outcomes, providing their asymptotic distributions and confidence bands under missing at random assumptions.

Findings

Both estimators have Gaussian process limiting distributions.

One estimator is double robust, requiring only one model to be correctly specified.

Simulations confirm good finite sample performance.

Abstract

We present and study semi-parametric estimators for the mean of functional outcomes in situations where some of these outcomes are missing and covariate information is available on all units. Assuming that the missingness mechanism depends only on the covariates (missing at random assumption), we present two estimators for the functional mean parameter, using working models for the functional outcome given the covariates, and the probability of missingness given the covariates. We contribute by establishing that both these estimators have Gaussian processes as limiting distributions and explicitly give their covariance functions. One of the estimators is double robust in the sense that the limiting distribution holds whenever at least one of the nuisance models is correctly specified. These results allow us to present simultaneous confidence bands for the mean function with asymptotically guaranteed coverage. A Monte Carlo study shows the finite sample properties of the proposed functional estimators and their associated simultaneous inference. The use of the method is illustrated in an application where the mean of counterfactual outcomes is targeted.