Estimation beyond Missing (Completely) at Random

This paper, titled "Estimation Beyond Missing (Completely) At Random," presents a theoretical framework for statistical inference when data is subject to missingness mechanisms that violate the standard Missing Completely At Random (MCAR) assumption.

Below is a detailed technical summary of the work.

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1. The Problem: The Failure of MCAR and MAR

In classical statistics, the MCAR assumption implies that the mechanism causing data to be missing is independent of both the observed and unobserved data. While mathematically convenient, MCAR is often unrealistic in practice (e.g., in medical studies where sicker patients are less likely to respond, or in econometrics where high-income individuals underreport earnings).

The most common alternative is Missing At Random (MAR), where missingness depends only on observed variables. However, the authors highlight two critical flaws in relying on MAR:

1. Identifiability Issues: Even in simple parametric models, MAR may not be sufficient to identify population parameters (e.g., the population mean).
2. Lack of Robustness: MAR does not provide a continuous way to measure how "close" a distribution is to the ideal MCAR case, making it difficult to quantify the risk of model misspecification.

2. Methodology: Huber-Style Contamination Models

To bridge the gap between the restrictive MCAR and the highly general Missing Not At Random (MNAR), the authors introduce two new classes of models inspired by Huber’s robust statistics:

• Arbitrary $\epsilon$-Contamination Model ($P_{arb}$): This model treats missingness as a mixture. With probability $(1-\epsilon)$, the data follows an MCAR mechanism; with probability $\epsilon$, the data follows an arbitrary, potentially adversarial, MNAR distribution. This is a "worst-case" framework.
• Realisable $\epsilon$-Contamination Model ($\mathcal{R}$): This is the paper's primary innovation. It assumes that the "contaminated" part of the data is not arbitrary, but is instead an MNAR version of the same base distribution $P$ that generates the uncontaminated part. This captures the reality that all data (observed or not) usually stems from a single underlying population.

The authors utilize a minimax quantile framework to evaluate estimator performance. This is necessary because, under these models, the standard minimax risk (expected error) is often infinite due to the possibility of observing zero data points.

3. Key Contributions and Results

A. Multivariate Mean Estimation (Arbitrary Contamination)

The authors propose the ITERATIVE_ROBUST_MEAN algorithm, which uses iterative imputation to convert a robust estimator for complete data into one for missing data.

• Result: They prove that the minimax error decomposes into an MCAR term (the error one would expect if data were missing completely at random) plus a robust error term that scales with $\epsilon$.
• Optimality: They show this algorithm is minimax optimal for the arbitrary contamination class.

B. The Power of Realisability (Gaussian and Nonparametric)

The paper demonstrates that if we assume the contamination is "realisable" (from the same base distribution), we can achieve much better rates than in the arbitrary case.

• Gaussian Case: For a Gaussian base distribution, they introduce a Minimum Kolmogorov Distance Estimator.
• Result: Unlike the arbitrary model (where estimation becomes impossible if $\epsilon$ is too large), in the realisable model, consistent estimation is possible even as the proportion of missingness $\epsilon$ approaches 1, provided it converges slowly enough.
• Nonparametric Case: They extend these results to distributions with only finite moments or specific tail decay (Orlicz norms), showing that the error scales with the effective contamination level $\kappa = \epsilon / (q(1-\epsilon))$.

C. Extension to Linear Regression

The authors extend their framework to normal linear regression where the response variable $Y$ is missing.

• Result: They prove that even with realisable missing responses, one can consistently estimate the regression coefficient vector $\theta$, provided the design matrix satisfies a "regularity" condition (ensuring the covariates are sufficiently spread out).

D. Adaptation to Unknown $\epsilon$

The authors provide a Lepski-type procedure that allows a practitioner to estimate population parameters without knowing the exact contamination proportion $\epsilon$ in advance. They prove that the "price of adaptation" (the loss in efficiency for not knowing $\epsilon$) is negligible.

4. Summary of Minimax Rates (Key Findings)

The paper's technical core is summarized in the comparison of rates (Table 1 in the text):

Base Distribution	Arbitrary Contamination Rate	Realisable Contamination Rate
Gaussian	$\infty$ if $\epsilon \geq \frac{q}{1+q}$	Consistent even as $\epsilon \to 1$
Sub-Gaussian	$\infty$ if $\epsilon \geq \frac{q}{1+q}$	Consistent even as $\epsilon \to 1$
Finite Variance	$\infty$ if $\epsilon \geq \frac{q}{1+q}$	Consistent even as $\epsilon \to 1$

5. Significance

This paper is significant because it provides a rigorous mathematical foundation for sensitivity analysis in missing data problems. By moving away from the binary "is it MAR or not?" question and toward a continuous "how much does it depart from MCAR?" framework, it allows statisticians to:

1. Quantify the impact of non-random missingness on their estimates.
2. Develop estimators that are robust to adversarial missingness patterns.
3. Achieve high-performance inference in settings previously thought to be unidentifiable.