Doubly robust estimation with functional outcomes missing at random

1. Problem Statement

The paper addresses the statistical challenge of estimating the mean of functional outcomes (random elements in $L^2([0, 1], \mathbb{R})$) when data is Missing at Random (MAR).

• Context: In many studies (e.g., lifetime income trajectories, human movement analysis), the outcome is a continuous curve rather than a scalar. However, for some units, the entire curve is missing due to non-response, drop-out, or counterfactual scenarios in causal inference.
• Assumption: The missingness mechanism depends only on observed baseline covariates ($X$), satisfying the MAR assumption: $Pr(Z=1|X, Y) = Pr(Z=1|X)$, where $Z$ is the indicator of observation.
• Gap: While semi-parametric estimators (Outcome Regression and Doubly Robust) are well-established for scalar outcomes, their extension to infinite-dimensional functional data is underdeveloped. Existing literature on fragmented functional data typically focuses on reconstructing missing parts of a curve using only observed curves, ignoring covariates. This paper fills the gap by utilizing covariate information to estimate the mean of the entire missing function.

2. Methodology

2.1 Model Setup

The authors define a working model for the functional outcome $Y_i$ given covariates $x_i$:
$$Y_i = x_i^\top \beta + \epsilon_i$$
where $\beta$ is a vector of non-stochastic functions in $L^2$, and $\epsilon_i$ is a zero-mean random element with covariance function $\sigma_\epsilon$.
The missingness probability is modeled via logistic regression:
$$Pr(Z_i=1|x_i) = \tau(x_i^\top \gamma) = (1 + e^{-x_i^\top \gamma})^{-1}$$

2.2 Estimators

Two semi-parametric estimators for the functional mean $\mu_y = E(Y)$ are proposed:

1. Outcome Regression (OR) Estimator:
   Based solely on the outcome model (1). It imputes missing outcomes using the regression prediction:
   $$\hat{\mu}_{OR} = n^{-1} \sum_{i=1}^n x_i^\top \hat{\beta}$$
   where $\hat{\beta}$ is the functional least squares estimator.

2. Doubly Robust (DR) Estimator:
   Combines the outcome model and the propensity score model (2). It corrects the OR estimator using inverse probability weighting for the residuals:
   $$\hat{\mu}_{DR} = n^{-1} \sum_{i=1}^n \left( x_i^\top \hat{\beta} + \frac{Z_i}{\tau(x_i^\top \hat{\gamma})} (Y_i - x_i^\top \hat{\beta}) \right)$$
   where $\hat{\gamma}$ is the MLE for the missingness model.

2.3 Asymptotic Theory

The core theoretical contribution involves establishing the limiting distribution of these estimators as Gaussian Processes (GPs).

• Strategy: The authors use a two-step approach based on Prohorov's Theorem:
  1. Finite-dimensional convergence: They discretize the domain $[0,1]$ into $T$ points, showing the resulting multivariate estimators converge to a multivariate normal distribution. They explicitly derive the asymptotic covariance matrices for these finite-dimensional vectors.
  2. Tightness: They prove the tightness of the sequence of probability measures in the Hilbert space to extend the finite-dimensional results to the infinite-dimensional functional setting.
• Key Result: The scaled error $\sqrt{n}(\hat{\mu} - \mu_y)$ converges weakly to a Gaussian process with mean zero and a specific covariance function.

3. Key Contributions

1. Limiting Distributions for Functional Estimators: The paper provides the first explicit characterization of the asymptotic covariance functions for both OR and DR estimators in the functional data context.
2. Double Robustness in Functional Settings: It proves that the functional DR estimator retains the "double robustness" property: the estimator is consistent and asymptotically normal (converging to a GP) if either the outcome regression model or the missingness mechanism model is correctly specified.
3. Simultaneous Confidence Bands (SCB): Leveraging the derived Gaussian process limits, the authors construct Simultaneous Confidence Bands for the entire mean function. Unlike pointwise intervals, these bands guarantee coverage over the entire domain $[0,1]$ with a specified confidence level (e.g., 95%). They utilize the "fast and fair" method (Liebl & Reimherr, 2023) to compute adaptive critical value functions.
4. Causal Inference Application: The framework is applied to a counterfactual question: estimating the lifetime income trajectory of a cohort had they all lived in large labor markets, utilizing the potential outcomes framework.

4. Results

4.1 Monte Carlo Simulation

The authors conducted extensive simulations with sample sizes $n \in \{250, 500, 1000, 3000\}$ and two error structures: Gaussian and Multivariate $t$ (to test robustness to non-smoothness).

• Coverage:
  • When models are correctly specified, the Simultaneous Confidence Bands (SCB) achieved empirical coverage close to or above the nominal 95% level.
  • Pointwise Confidence Bands (PCB) performed poorly (coverage dropped significantly as $n$ increased) because they fail to account for the multiplicity of comparisons across the function domain.
  • The DR estimator maintained correct coverage even when one of the two working models was misspecified.
• Bias and Efficiency:
  • The DR estimator showed negligible bias when at least one model was correct.
  • When both models were misspecified, bias appeared, but the DR estimator generally outperformed the naive "complete case" estimator (which suffered from large bias due to ~30% missing data).
  • The method remained robust even when the smoothness assumption (once continuously differentiable) was violated by using $t$-distributed errors, though coverage was slightly lower.

4.2 Real Data Application

The method was applied to Swedish cohort data (born 1954) to estimate the counterfactual lifetime income trajectory if all individuals had lived in large labor markets at age 20.

• Finding: The naive estimator (using only observed data from large markets) overestimated early income but underestimated accumulated income before age 30 compared to the DR estimate.
• Inference: The simultaneous confidence bands allowed the authors to identify specific age intervals where the difference between the counterfactual and observed trajectories was statistically significant.

5. Significance and Conclusion

This paper bridges a critical gap between missing data methodology and functional data analysis.

• Theoretical Impact: It extends the well-known double robustness property from scalar to functional outcomes, providing rigorous asymptotic theory (Gaussian process limits) and explicit covariance structures.
• Practical Utility: By enabling the construction of simultaneous confidence bands, the method allows researchers to make valid global inferences about functional means (e.g., "Is the trajectory significantly different between groups for the entire duration?") rather than just pointwise inferences.
• Robustness: The DR estimator offers a safeguard against model misspecification, which is crucial in observational studies where the true data-generating process is unknown.

The authors conclude that their approach provides a reliable tool for causal inference and mean estimation in functional data settings with missing values, offering both asymptotic guarantees and satisfactory finite-sample performance.