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Asymptotics of Nonparametric Estimation under General Non-monotone MAR Missingness: A Nonparametric Maximum Likelihood Approach

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that under general non-monotone missing at random with a positivity condition, a sieve maximum-likelihood estimator recovers the complete-data density at the minimax rate up to a logarithmic factor, with missingness…

desk verdict A genuine first: frequentist sieve MLE rates under general non-monotone MAR, with a load-bearing positivity assumption that is honestly disclosed. read the letter →

arxiv 2608.10113 v1 pith:FBC5RM43 submitted 2026-08-10 stat.ME

Yating Zou , Huimin Hu , Jeffrey Näf This is my paper · ORCID
classification stat.ME MSC 62G0762G2062D10
keywords missingatrandomnon-monotonemissingnesssievemaximumlikelihoodnonparametricdensityestimationminimaxrateGaussianmixtureadaptedHellingerfunctionalpositivitycondition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Missing at random (MAR) promises that the missingness mechanism can be ignored when maximizing likelihood, but for non-monotone missingness—where each variable can be missing in arbitrary combinations—frequentist guarantees for nonparametric estimation have been scarce. This paper establishes that a sieve maximum-likelihood estimator, maximizing only the observed-data marginal likelihood over a growing space of Gaussian mixtures, recovers the complete-data density consistently under general MAR, with no model for the missingness mechanism and no restriction on the configuration of missingness patterns. The rate is the minimax rate for a $\beta$-smooth density up to a logarithmic factor, and missingness enters only through a constant $1/\sqrt{c_0}$ arising from the positivity condition. This matters because it turns the long-held intuition that likelihood estimation ignores MAR into a theorem, while also providing a practical EM algorithm that performs like a complete-data kernel density estimator.

What carries the argument

The load-bearing object is the adapted Hellinger functional, defined by summing pattern-wise squared Hellinger differences between observed-block marginals, weighted by $P(M=m|x^{(m)})$: $$\widetilde{H}^2(f,g)=\sum_m \int \big(\sqrt{$f^{{(m)}}$($x^{{(m)}}$)}-\sqrt{$g^{{(m)}}$($x^{{(m)}}$)}\big)^2\, P(M=m|$x^{{(m)}}$)\,$dx^{{(m)}}$.$$ Under positivity it behaves like a distance and is squeezed between $c_0 H^2$ and $H^2$, so convergence in it implies convergence of the complete density and bracketing-entropy bounds transfer from complete data to observed data. The proof also uses a Gaussian-mixture sieve whose covariance eigenvalues are allowed to diverge with $n$, a localization argument that caps the eigenvalues for densities near the truth, and recent Hellinger bounds on the Kullback-Leibler divergence and the Bernstein norm that remove the need for bounded density ratios.

What would settle it

Take a true density $p_0$ and a missingness mechanism with $P(M=0|X=x)=0$ on a cube of positive Lebesgue measure, and choose $p_1$ equal to $p_0$ outside that cube but different inside. The observed-data likelihood is identical under $p_0$ and $p_1$, so no sieve MLE can converge to the true density in Hellinger distance on that region; a simulation with such a mechanism would show the estimated density failing to recover $p_0$ exactly where the chance of full observation is zero.

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Extended reading notes

Core claim

The central discovery is Theorem 5.1: under MAR, a positivity condition $P(M=0|X=x)\ge c_0>0$, and a H\"older smoothness condition with exponent $\beta$, any approximate maximizer $\hat p_n$ of the observed-data likelihood over the Gaussian-mixture sieve satisfies $H(p_0,\hat p_n)=O_P\big((C_4/\sqrt{c_0})\, n^{-\beta/(2\beta+d)}(\log n)^t\big)$. Thus the complete-data density is estimated at the minimax rate up to a logarithmic factor, for any prescribed smoothness level, even when each coordinate can be missing in arbitrary non-monotone patterns. The missingness mechanism is never modeled or estimated; it affects only the multiplicative constant through $c_0$. Theorem 4.4 supplies the general sieve-MLE rate under MAR that powers this density result, and the paper shows the estimator is approximated in practice by a simple EM algorithm operating directly on the incomplete data.

Load-bearing premise

Every data point must have at least a fixed positive chance of being observed completely; if some region of the covariate space never produces a fully observed record, the observed data cannot tell apart densities that differ only there, and the claimed rate fails.

Editorial extensions

If this is right

  • Ignoring maximum likelihood is now a valid nonparametric estimator under general non-monotone MAR: consistency and rates hold without specifying a model for why values are missing.
  • The rate is minimax up to logarithmic factors: additional missingness patterns, or complex dependencies between missingness and observed values, do not slow the density estimator beyond the constant factor $1/\sqrt{c_0}$.
  • Any statistic that is a continuous functional of the density—such as quantiles or smooth M-estimators—inherits consistency when applied to the estimated density.
  • The estimator is computable: a closed-form EM update on the incomplete data approximates the sieve maximizer, so the theoretical guarantee has a practical algorithm attached.
  • Complete-data kernel density estimators with Gaussian kernels are limited to smoothness $\beta\le 2$, while the Gaussian-mixture sieve reaches near-minimax rates for every $\beta>0$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If positivity fails on a region of positive measure, the adapted Hellinger functional ceases to identify the complete density there; pattern-specific identification assumptions would be needed, so the positivity condition marks the boundary of what MAR alone identifies.
  • The same proof architecture should transfer to other sieves, such as nonparametric maximum likelihood for Gaussian location mixtures or sieve estimators in nonparametric regression, making Theorem 4.4 a template for ignoring estimators beyond density estimation.
  • A practical diagnostic extension would estimate the chance of full observation and flag samples where its minimum is near zero, since the rate constant $1/\sqrt{c_0}$ warns that estimation will be poor in low-observation regions even when consistency holds.
  • The logarithmic factor originates in the entropy of the Gaussian-mixture sieve rather than in the missingness, so removing it may be possible with sharper entropy or localization arguments.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a frequentist sieve maximum likelihood theory for estimation under non-monotone missing at random (MAR) missingness. It introduces an adapted Kullback-Leibler divergence and an adapted Hellinger functional on observed-data marginals, and proves (Theorem 4.4) that an approximate maximizer of the observed-data likelihood over any sieve satisfying bracketing entropy, approximation, and moment conditions converges at rate eH(p0, p_hat n)=O_P(delta_n or n^{-1/2}), with H-rate degraded by a 1/sqrt(c0) factor under a uniform positivity assumption on the propensity score. Applying this to a Gaussian mixture sieve under a Holder smoothness class with exponential tails (Theorem 5.1), it obtains H(p0, p_hat n)=O_P(C4/sqrt(c0) n^{-beta/(2beta+d)} (log n)^t), the minimax rate up to logarithmic factors, with missingness affecting only the constant. The estimator is implemented by a MAR-specific EM algorithm with BIC/AIC selection; simulations in d=20 compare it with complete-data Gaussian KDE. The proofs are detailed in an appendix, although two key identifiability and rate-bridge propositions are cited from an unpublished companion paper.

Significance. If correct, this is a substantial contribution: it provides the first general frequentist rate guarantees for nonparametric likelihood-based estimation under non-monotone MAR, and it removes the bounded-density-ratio condition via Kaji's Hellinger bounds. The adapted-divergence machinery is clean, and the adapted Hellinger functional is well suited to pattern-marginal data. The result that missingness changes only the constant in the minimax rate is striking and is stated with transparent assumptions. The authors are also candid about the role of the positivity assumption, and the new entropy localization arguments in Lemmas B.6-B.9 are worked out in detail. Reproducible code is provided. The significance is moderated, however, by the reliance on a strong uniform positivity condition, by the restriction to Holder classes with strong tail and moment conditions and known smoothness, and by the fact that two load-bearing results (Propositions 4.1 and 4.3) are borrowed from an unpublished companion preprint rather than proved in the manuscript.

major comments (3)
  1. [Section 4, Propositions 4.1 and 4.3] The identifiability of the adapted KL divergence (Proposition 4.1) and the two-sided bound c0 H^2(p1,p2) <= eH^2(p1,p2) <= H^2(p1,p2) (Proposition 4.3) are both cited from the companion preprint Naf and Cherief-Abdellatif (2026), which is not peer-reviewed. These propositions are load-bearing: Proposition 4.1 justifies the target of the sieve MLE, and Proposition 4.3 is used in Lemma B.3 and in the final transition from eH to H in Theorems 4.4 and 5.1. For a self-contained journal submission, the full proofs of these propositions should be included in the appendix or derived in the paper, rather than referenced to an unpublished source.
  2. [Section A.1, Algorithm 1, lines 19-22] The quantity ell_k is computed as sum_i sum_v (1-m_iv) phi(...) + log pi_k, and then ell = log sum_k exp(ell_k). As written, this is not the observed-data log-likelihood: the component contribution should use log phi rather than phi, and the sum over observations should be outside the log-sum-exp, i.e. ell = sum_i log sum_k exp(log pi_k + sum_v (1-m_iv) log phi(...)). The convergence check and the BIC/AIC criterion therefore use an incorrect objective, which can affect the simulation results and model selection; the pseudocode should be corrected and the simulations should be verified with the corrected likelihood.
  3. [Section 5 and Section 6] Theorem 5.1 is stated for the sieve P_n with general, possibly non-diagonal covariance matrices, but the implemented estimator restricts to diagonal covariances, sieve fP_n. The paper asserts that the same asymptotic result could be shown for fP_n, but no proof is given. Since fP_n subset P_n does not imply that an approximate maximizer over fP_n is an approximate maximizer over P_n in the sense of condition (7'), the theorem does not formally cover the estimator used in the simulations. Please extend Theorem 5.1 to the diagonal sieve or provide the missing argument.
minor comments (4)
  1. [Abstract and Section 1] The claim that the estimator attains the minimax rate 'for any prescribed smoothness level' should be qualified: Theorem 5.1 requires Assumption 5.1 with known parameters beta and tau, and the sieve tuning in (11) depends on beta. The result is a fixed-smoothness guarantee, not an adaptive one.
  2. [Section 2, Assumption 2.3 and Remark 1] Assumption 2.3 is not implied by MAR, and the paper itself notes in Section 4 that without pointwise positivity the adapted KL functional no longer identifies P0. The phrase 'natural positivity condition' in the abstract understates this: the rate constant 1/sqrt(c0) can be arbitrarily large. A short discussion of when c0 can be expected to be bounded away from zero, and of the consequences of small c0, would help the reader calibrate the scope of the result.
  3. [Algorithm 2, line 8] The responsibility update in Algorithm 2 has a typo: the denominator uses pi_j p_{i ell} where it should be pi_ell p_{i ell}. The same typo appears in the E-step exposition following equation (15).
  4. [Figures 2 and 3] The captions refer to colors and dashed lines (e.g., 'dashed blue line'), but the figures appear without legends or labeled curves. Please add direct labels or a legend so the results are interpretable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rate theorem is derived by verifying sieve entropy and bias conditions, and the self-citations transmit independent lemmas rather than the target conclusion.

full rationale

The paper's central claim, Theorem 5.1, states that an approximate maximizer of the observed-data likelihood over a Gaussian-mixture sieve satisfies H(p0, p_hat_n) = O_P((C4/sqrt(c0)) n^{-beta/(2beta+d)} (log n)^t). This conclusion is obtained by checking the three conditions (4'), (5'), and (6') of the general sieve-MLE rate theorem, Theorem 4.4, on a concrete sieve; the proof in Appendix B supplies the entropy and approximation arguments rather than assuming the rate. The quantity delta_n is a candidate rate verified through bracketing entropy bounds and a bias-correction construction, not a parameter fitted to the conclusion. Missingness enters only through the constant c0, via Proposition 4.3, whose bound c0 H^2(p1,p2) <= eH^2(p1,p2) <= H^2(p1,p2) is an immediate consequence of the definition of the adapted Hellinger functional and Assumption 2.3: the m=0 pattern contributes H^2(p1,p2) with weight P(M=0|x) >= c0, and the upper bound follows from Cauchy-Schwarz and summing propensity weights to one. Although Proposition 4.3 and Proposition 4.1 are cited from the companion paper by Naf and Cherief-Abdellatif (2026), they are parameter-free mathematical lemmas whose assumptions do not include the frequentist rate theorem being proved, so they are independent support rather than a self-citation chain. The approximate-maximizer condition (7') is the standard condition from the external sieve-MLE theory of Kaji and van der Vaart and Wellner, and the paper verifies it in the proof. There is no fitted input renamed as a prediction, no uniqueness theorem imported to forbid alternatives, and no ansatz smuggled in via citation. Assumption 2.3 is indeed load-bearing, but the abstract and Remark 1 explicitly present it as a stated positivity condition; a scope limitation is not circularity. The simulation study is empirical and does not enter the proof chain. Therefore the derivation does not reduce to its own inputs and no circular step is present.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard missing data assumptions (MAR, positivity, no empty pattern) plus a Holder class condition with tail restrictions. The sieve parameters C4 and t are theoretical constants, not fitted to data. No new physical or structural entities are postulated.

free parameters (3)
  • C4
    Large enough constant in the rate delta_n = C4 n^{-beta/(2beta+d)} (log n)^t; chosen to satisfy entropy and bias conditions in the proof, not fitted to data.
  • t
    Exponent in the logarithmic factor, required to satisfy inequality (10); a theoretical constant chosen in the proof, not fitted.
  • Covariance regularization lambda
    Regularization added to covariance updates in Algorithm 1 to ensure positivity; chosen by the user in practice, not part of the main theorem.
assumptions (5)
  • domain assumption Assumption 2.1 (MAR): P(M=m|X=x) = P(M=m|X^(m)=x^(m)) for all m, x.
    Core missingness assumption; used throughout to rewrite observed-data likelihood as a propensity-weighted marginal expectation.
  • domain assumption Assumption 2.2: P(M=1)=0 (the completely missing pattern has probability zero).
    Avoids observing no coordinates; needed for the observed-data likelihood to be well-defined.
  • domain assumption Assumption 2.3: P(M=0|X=x) >= c0 > 0.
    Positivity of the propensity score; turns the adapted Hellinger pseudometric into a metric and sets the constant in the rate.
  • domain assumption Assumption 5.1: Holder class with moment conditions and exponential tail.
    Smoothness and tail assumptions on the true density p0 needed for Gaussian mixture approximation and entropy bounds.
  • standard math Empirical process theory: van der Vaart and Wellner (2023) Theorem 3.4.1 and Kaji (2026) inequalities.
    Framework for sieve MLE rates; the paper adapts these tools to the MAR setting.

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Pith. "Pith review of Asymptotics of Nonparametric Estimation under General Non-monotone MAR Missingness: A Nonparametric Maximum Likelihood Approach." pith.science (2026). https://pith.science/paper/FBC5RM43

@misc{pith2026260810113,
  author       = {Pith},
  title        = {Pith review of: Asymptotics of Nonparametric Estimation under General Non-monotone MAR Missingness: A Nonparametric Maximum Likelihood Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBC5RM43}},
  note         = {Machine review of arXiv:2608.10113}
}
read the original abstract

Missing data constitute a pervasive challenge in empirical research. Consequently, there is an ever-growing number of methods designed to address this challenge, with multiple imputation and inverse probability weighting the dominant strategies. Despite this, theoretical guarantees remain limited, particularly in the challenging case of non-monotone missing at random (MAR). When guarantees exist, they are often confined to simplified settings such as missing completely at random, monotone or block-wise missingness, or rest on restrictive assumptions about the missingness mechanism. In this paper, we utilize the theory of sieve maximum likelihood to establish a general rate of convergence under MAR that requires no modeling of the missingness mechanism and no restriction on the configuration of missing patterns, beyond MAR itself and a natural positivity condition. Applying this result to density estimation, we show that the complete-data density can be estimated at the minimax rate over a H\"older class, up to a logarithmic factor, for any prescribed smoothness level. The missingness does not affect the rate and enters only through a constant. The estimator is approximated in practice by a simple expectation-maximization (EM) algorithm operating on the incomplete data directly. In simulations, it performs comparably to the kernel density estimator supplied with the complete data across a wide range of missingness levels.

Figures

Figures reproduced from arXiv: 2608.10113 by the authors.

Figure 1
Figure 1. Three Data matrices with missing values, each with three different patterns. Each [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Estimation of the 0.1 quantile of X1 (left) and correlation between X1, X2 (right), for B = 50, with n = 5000 observations each. The dashed blue line corresponds to the correct value. “missForest” refers to the widely-used method of Stekhoven and B¨uhlmann (2011), “mean” to mean imputation, where missing values are replaced by the sample mean, “MLE” to the ignoring MLE in (1) and “Mest” to the ignoring M-estimator, … view at source ↗
Figure 3
Figure 3. Estimation of 0.1 quantile of X1 for data in the motivating example, with B = 50 and n = 5000 observations for each batch. G(x) = max(Φ(x), c0) to ensure P(M = m1 | x) ≥ c0. Capability in recovering complete-data density for d = 20. Here, we compare our method fitted on data with varying levels of missingness, against the Gaussian Kernel Density Estimator (KDE) with complete data. We directly apply scipy.stats.gauss… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Data generated from a 20-component Gaussian Mixture with randomly simulated loca [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: shows that our method outperforms Gaussian KDE at all missingness levels in terms of energy distance. This implies that the point estimate from our method is very close to the truth on a global scale since the energy distance measures the overall closeness between two …
Figure 6
Figure 6. Figure 6: Multivariate logistic data consists of variables with standard logistic marginal distribu [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Data sampled from N20(0, Σ) with X1, X2, . . . , X19 correlated, and X20 independent from others. For our method, we use shared diagonal covariances for all components of base models, and select the model by the AIC criterion. The number of random initializations is 30…
Figure 8
Figure 8. Figure 8: Multivariate logistic data consists of variables with standard logistic marginal distribu [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

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