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arxiv: 2410.10704 · v2 · submitted 2024-10-14 · 🧮 math.ST · stat.ME· stat.TH

Estimation beyond Missing (Completely) at Random

Tianyi Ma , Kabir A. Verchand , Thomas B. Berrett , Tengyao Wang , Richard J. Samworth This is my paper

Pith reviewed 2026-05-23 19:15 UTC · model grok-4.3

classification 🧮 math.ST stat.MEstat.TH
keywords mean estimationmissing dataepsilon-contaminationMCARMNARminimax riskrobust estimation
0
0 comments X

The pith

Minimax mean estimation error under arbitrary missingness decomposes into MCAR quantiles plus an epsilon-dependent robust term.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper formulates a missing-data version of Huber's arbitrary epsilon-contamination model. It proves that, for mean estimation under squared Euclidean loss, the minimax quantiles in this model equal the sum of the minimax quantiles under a heterogeneous MCAR model and an additional robust error term that depends only on epsilon. This separation quantifies how much extra error arises once missingness departs from MCAR. The authors further introduce realisable epsilon-contamination classes, formed by contaminating an MCAR version of a base distribution P with an arbitrary MNAR version of the same P; these classes still capture biased sampling yet yield strictly better minimax rates than the arbitrary classes for both parametric and nonparametric base distributions.

Core claim

For mean estimation under squared Euclidean loss, the minimax quantiles under the arbitrary epsilon-contamination model decompose as the sum of the corresponding minimax quantiles under a heterogeneous MCAR assumption and a robust error term depending on epsilon.

What carries the argument

The realisable epsilon-contamination classes, where an MCAR version of base distribution P is contaminated by an arbitrary MNAR version of P.

If this is right

  • Consistent mean estimation remains possible over the realisable classes even when both epsilon and the missingness proportion converge slowly to 1, for a univariate Gaussian base distribution.
  • The decomposition and rate improvements extend to departures from MAR in normal linear regression when the missing response follows a realisable model.
  • The procedures can be made adaptive to the case of unknown epsilon.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decomposition suggests constructing estimators by solving the MCAR problem first and then adding a separate robust correction whose size is controlled by epsilon.
  • The same realisable-class construction could be tested on other functionals such as quantiles or covariance matrices to see whether the additive separation persists.
  • In practice, one could check whether observed data patterns are consistent with the realisable model by verifying that the complete cases and the missingness mechanism can be generated from a single base distribution P.

Load-bearing premise

The realisable epsilon-contamination classes capture biased sampling and sensitivity conditions while still permitting the improved minimax performance stated for both parametric and nonparametric base distributions.

What would settle it

A concrete base distribution and sequence of epsilon values where the observed minimax quantile under arbitrary contamination fails to equal the sum of the heterogeneous MCAR quantile and the claimed robust term.

Figures

Figures reproduced from arXiv: 2410.10704 by Kabir A. Verchand, Richard J. Samworth, Tengyao Wang, Thomas B. Berrett, Tianyi Ma.

Figure 1
Figure 1. Figure 1: An illustration of the arbitrary ϵ-contamination model P arb(P, ϵ, π), which interpolates between MCAR(π,P) and P(X⋆). 2.2.2 Huber-style models of departure from MCAR Given the failure of the MAR assumption to ensure the tractability of the mean estimation problem, and in light of dual representation of the incompatibility index given by Berrett and Samworth (2023, Theorem 2), it is natural to model depart… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of the realisable ϵ-contamination model R(P, ϵ, π), which interpolates between MCAR(π,P) and MNARP . the realisable contamination model R(P, ϵ, π) represents a (still nonparametric) subclass of P arb(P, ϵ, π), with the potential to yield improved rates of mean estimation. On the other hand, noting that R(P, 1, π) = MNARP , in Example 2, the distribution Rθ belongs to MNARP but not to R(P, ϵ… view at source ↗
Figure 3
Figure 3. Figure 3: An example of a Gaussian-realisable distribution. Let [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the Kolmogorov projection onto two distinct realisable sets. The realisable [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic diagrams of various maps defined in the proof. The fact that the maps in the [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Construction of the lower bound in Theorem [PITH_FULL_IMAGE:figures/full_fig_p047_6.png] view at source ↗
read the original abstract

We study the effects of missingness on the estimation of population parameters. Moving beyond restrictive missing completely at random (MCAR) assumptions, we first formulate a missing data analogue of Huber's arbitrary $\epsilon$-contamination model. For mean estimation with respect to squared Euclidean error loss, we show that the minimax quantiles decompose as a sum of the corresponding minimax quantiles under a heterogeneous, MCAR assumption, and a robust error term, depending on $\epsilon$, that reflects the additional error incurred by departure from MCAR. We next introduce natural classes of realisable $\epsilon$-contamination models, where an MCAR version of a base distribution $P$ is contaminated by an arbitrary missing not at random (MNAR) version of $P$. These classes are rich enough to capture various notions of biased sampling and sensitivity conditions, yet we show that they enjoy improved minimax performance relative to our earlier arbitrary contamination classes for both parametric and nonparametric classes of base distributions. For instance, with a univariate Gaussian base distribution, consistent mean estimation over realisable $\epsilon$-contamination classes is possible even when $\epsilon$ and the proportion of missingness converge (slowly) to 1. We extend our results to the setting of departures from missing at random (MAR) in normal linear regression with a realisable missing response, and also demonstrate that our methods can be made adaptive to the case of unknown $\epsilon$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript formulates a missing-data analogue of Huber's arbitrary ε-contamination model. For mean estimation under squared Euclidean loss it establishes that the minimax quantiles decompose exactly into the sum of the corresponding minimax quantiles under a heterogeneous MCAR model plus an additive robust term that depends only on ε. It then defines realisable ε-contamination classes in which an MCAR version of a base distribution P is contaminated by an arbitrary MNAR version of the same P; these classes are shown to admit strictly better minimax rates than the arbitrary contamination model for both parametric and nonparametric base distributions. A concrete illustration is given for univariate Gaussians, where consistent estimation remains possible even when both ε and the missingness proportion converge slowly to 1. The results are extended to departures from MAR in normal linear regression with realisable missing responses and to the case of unknown ε.

Significance. If the decomposition and the rate improvements for the realisable classes hold, the work supplies a precise quantitative separation between the cost of missingness under MCAR and the additional cost incurred by MNAR departures. The realisable classes identify a nontrivial intermediate regime between fully arbitrary contamination and classical MCAR/MAR assumptions, and the Gaussian example demonstrates that consistent estimation can survive regimes previously thought intractable. The regression extension and the adaptation result further increase the scope of the framework within theoretical statistics.

major comments (2)
  1. [Abstract] The central decomposition result is stated for squared Euclidean loss; the manuscript should verify whether the additivity continues to hold for other convex losses or whether the Euclidean geometry is essential (abstract, paragraph on the decomposition).
  2. [Abstract] The definition of the realisable ε-contamination classes must ensure that the MNAR contaminating measure is supported on the same base distribution P as the MCAR component; any hidden restriction on the support or on the missingness mechanism would affect the claimed improvement over arbitrary contamination (abstract, paragraph beginning 'We next introduce natural classes...').
minor comments (2)
  1. Notation for the heterogeneous MCAR model and the robust error term should be introduced with explicit symbols before the decomposition statement is used in later sections.
  2. The transition from the arbitrary contamination model to the realisable classes would benefit from a short motivating paragraph that contrasts the two classes with a simple numerical example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] The central decomposition result is stated for squared Euclidean loss; the manuscript should verify whether the additivity continues to hold for other convex losses or whether the Euclidean geometry is essential (abstract, paragraph on the decomposition).

    Authors: The decomposition is derived specifically for squared Euclidean loss; the proof exploits the Hilbert space geometry of L2 and the associated Pythagorean identity for projections, which does not extend verbatim to general convex losses. We will revise the abstract to state explicitly that the additivity holds for squared Euclidean loss and to note that the Euclidean structure appears essential. No claim of generality to other losses is made in the manuscript. revision: yes

  2. Referee: [Abstract] The definition of the realisable ε-contamination classes must ensure that the MNAR contaminating measure is supported on the same base distribution P as the MCAR component; any hidden restriction on the support or on the missingness mechanism would affect the claimed improvement over arbitrary contamination (abstract, paragraph beginning 'We next introduce natural classes...').

    Authors: The realisable classes are defined precisely so that both the MCAR and MNAR components are versions of the identical base distribution P (see the formal definition in Section 2.2 and the abstract sentence beginning 'We next introduce natural classes...'). The MNAR component shares the marginal law P while permitting arbitrary dependence between the missingness indicator and the observations. This shared-P requirement is explicit and is what yields the strict improvement over arbitrary contamination; no hidden support restrictions are present. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives minimax quantile decompositions directly from the definitions of the arbitrary epsilon-contamination model and the realisable classes, expressing them as sums of an MCAR component plus an additive robust term. These are presented as first-principles results obtained from the model classes themselves, with no reduction of any claimed prediction or theorem to a fitted parameter, self-citation chain, or definitional equivalence. The abstract and described results contain no load-bearing self-citations, no ansatz smuggled via prior work, and no renaming of known patterns as new derivations. The central claims remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the existence of well-defined minimax quantiles under the newly introduced contamination classes and on the technical validity of the decomposition identity; no explicit free parameters or invented entities are named in the abstract.

axioms (1)
  • domain assumption Minimax quantiles exist and are finite for the mean estimation problem under the stated loss and contamination models.
    Invoked implicitly when the abstract asserts that the quantiles 'decompose as a sum'.

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