Efficient Estimation under Multiple Missing Patterns via Balancing Weights

Jianing Dong; Kwun Chuen Gary Chan; Raymond K. W. Wong

REVIEW 3 major objections 4 minor 28 references

Efficient Estimation under Multiple Missing Patterns via Balancing Weights

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that a sequentially balanced weighting estimator is semiparametrically efficient for parameters defined by moment equations under non-monotone, missing-not-at-random patterns.

desk verdict Solid extension of pattern-graph missing-data estimation; the efficiency bound looks right, but a real proof gap in Lemma H.8 means the efficiency claim is not yet fully supported. read the letter →

arxiv 2504.13467 v1 pith:UYUCAFK2 submitted 2025-04-18 stat.ME

Jianing Dong , Raymond K. W. Wong , Kwun Chuen Gary Chan This is my paper · ORCID

classification stat.ME MSC 62D1062F1262G05

keywords missingnotatrandomnon-monotonedatapatterngraphsbalancingweightssemiparametricefficiencyweightedestimatingequationspropensityoddscovariate

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

The paper addresses estimation of parameters defined by moment conditions, such as regression coefficients, when data have multiple non-monotone missingness patterns and missingness may depend on unobserved values. Its proposal is to reweight complete cases with weights built by sequentially balancing each missing pattern against the complete cases, then solve a weighted estimating equation. The authors derive the semiparametric efficiency bound for the parameter under the identifying assumptions encoded by a regular pattern graph and prove that the sequential balancing estimator attains this bound. As a result, the simple weighted estimator is asymptotically as precise as any regular estimator, including imputation-based or augmented procedures. Simulations illustrate that the sequential weights are more stable and give smaller bias and mean squared error than entropy-based or local balancing weights.

What carries the argument

The load-bearing device is a recursive factorization of propensity odds along paths in a pattern graph, a directed acyclic graph whose vertices are missingness patterns and whose edges encode the identifying assumption that a pattern's unobserved variables follow the same conditional distribution as a mixture over its parents. For a missingness pattern $r$, the odds of seeing pattern $r$ rather than the complete case is $Q_r(l)=O_r(l[r])\sum_{s\in \mathrm{Pa}(r)} Q_s(l)$, with $Q_{1^d}=1$, where $O_r(l[r])$ is the local odds of pattern $r$ versus its parents given observed variables. The sequential balancing loss $L_r\{O_r(l[r];\alpha_r),R\}=1_{R=1^d}O_r(l[r];\alpha_r)\hat Q_{\mathrm{Pa}(r)}(l)-1_{R=r}\log O_r(l[r];\alpha_r)$ is minimized at each pattern in increasing missingness order, which enforces empirical balance between pattern $r$ and reweighted complete cases and controls error accumulation along the factorization. The final weights $\hat w(L)=\sum_{r\in\mathcal R}\hat Q_r(L)$ are inserted into the weighted estimating equation.

What would settle it

Construct a data-generating process satisfying the pattern-graph identification assumptions but with one propensity odds $O_r(l[r])$ unbounded above, e.g., $P(R\in\mathrm{Pa}(r)|l[r])$ decaying exponentially in an observed covariate; then run the sequential balancing estimator on many replications and check whether the fitted weights develop heavy tails and whether the coverage of 95% confidence intervals degrades, which would refute the uniformity claim on which efficiency rests.

Watch

Extended reading notes

Core claim

Under the identifying assumptions of a regular pattern graph, the paper establishes that the estimator $\hat\theta_N$ obtained by solving $\frac{1}{N}\sum_{i: R_i=1^d} \hat w(L_i)\psi_{\theta}(L_i)=0$ with $\hat w(L)=\sum_{r\in\mathcal R}\hat Q_r(L)$ is consistent for $\theta_0$ and satisfies $N^{1/2}(\hat\theta_N-\theta_0)\to_d N(0,D_{\theta_0}^{-1}V_{\theta_0}D_{\theta_0}^{-\top})$, where $D_{\theta_0}$ is the derivative of the moment and $V_{\theta_0}$ is the variance of the efficient influence function. The paper shows this variance coincides with the asymptotic variance bound for all regular estimators derived in Theorem 4.2, so the estimator attains semiparametric efficiency.

Load-bearing premise

The entire consistency and efficiency argument rests on uniform overlap: for each missing pattern the true propensity odds $O_r(l[r])$ must be bounded away from zero and infinity on the support of $l[r]$, and the complete-case probability must also be bounded away from zero, so no pattern pairing becomes arbitrarily rare.

Editorial extensions

If this is right

The estimator reaches the same asymptotic variance lower bound as any regular estimator, so under the stated assumptions no imputation or augmented IPW strategy can improve first-order efficiency.
Because the sequential loss uses complete cases directly in every pattern's optimization, the weight estimators avoid extrapolating odds models to the complete-case region, which the paper argues removes a source of instability.
The method extends balancing-weight estimation from the single-parent CCMV assumption to general regular pattern graphs with multiple parents and to generalized mixture coefficients.
Simulations show reductions in bias and mean squared error relative to entropy and local balancing weights, including settings where the identifying pattern graph is misspecified.
A consistent sandwich variance estimator accompanies the estimator, so confidence intervals and hypothesis tests can be constructed in practice.

Reading between the lines

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

A natural extension beyond the paper is to causal effect estimation with multiple versions of treatment or noncompliance patterns, since target causal contrasts can be written as moment equations.
The efficiency result implies that when the overlap assumption holds, the choice between imputation and weighted estimation is second-order; practical gains would come from stability and computation, where the sequential weights may dominate.
A diagnostic extension: evaluate the empirical balance condition for each pattern's basis functions; large imbalance under the fitted weights would signal that either the pattern graph or the overlap assumption is untenable.
One could test the overlap assumption indirectly by monitoring the distribution of fitted weights on complete cases; weights diverging toward infinity for a non-negligible fraction of observations would flag a violation.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Read this one carefully before you put money on the efficiency claim. The paper is a serious extension of Chen (2022) and the local balancing work: it handles general moment equations, derives the semiparametric efficiency bound under pattern-graph MNAR assumptions, and proposes a sequential balancing estimator designed to avoid the extrapolation and error-accumulation problems of local weighting. The efficiency bound derivation in Appendix C is standard semiparametric theory done carefully for all three mixture-coefficient types, and I think it is correct. The sequential loss is a sensible construction, and the simulation does show the method beating entropy and local balancing in the graphs considered.

Now the soft spots. The proof of Theorem 4.3 depends on Lemma H.8, and the lemma's proof is not a proof. It states an inequality relating the balancing-error term S^r_{θ,3} to the penalized objective, says 'Notice that', and then jumps to the conclusion. That inequality is doing real work: it is exactly how the balancing property gives the op(1) rate. Without a derivation, or a reference to a known result that implies it, the influence-function decomposition in Theorem D.4 is not complete. This is not a minor typo; it is a missing argument at the hinge of the main theorem. I would not call the claim false—the result is plausible and likely true given the structure—but the manuscript as written does not establish it.

Two smaller issues. The overlap assumption (D.1.A) is bounded propensity odds uniformly; the authors call it standard, and it is, but it is not testable and can easily fail with small complete-case probabilities. That is a limitation to state clearly. And the simulation appears to be nearly correctly specified: the true odds are polynomial degree four and the basis functions are splines up to degree four, so the simulation is closer to an oracle check than to an honest evaluation under misspecification. Also, no code or data is provided, which makes it harder to check the finite-sample behavior. These are fixable in revision.

Verdict: worth refereeing. The efficiency bound alone is a contribution, and the estimator design is interesting enough that the missing lemma deserves a careful look, not a desk reject. Give it to a referee who knows the balancing literature and ask specifically about Lemma H.8.

Referee Report

3 major / 4 minor

Summary. This paper studies estimation of parameters defined by moment equations when data have multiple non-monotone missing patterns and missingness is not at random. Building on pattern graphs and the authors' prior work on balancing weights, it proposes a sequential balancing loss that estimates the propensity odds O_r sequentially from complete cases, constructs weights ŵ = Σ_r Q̂_r, and solves a weighted estimating equation. The paper derives a semiparametric efficiency bound (Theorem 4.2), claims consistency and semiparametric efficiency of the sequential estimator (Theorem 4.3), and presents simulations and a real-data application. The theoretical development relies on a long appendix with empirical-process and bracketing-number arguments, including sieve approximations and several technical lemmas.

Significance. The intended contribution is significant: if Theorem 4.3 were rigorously established, the paper would provide a computationally simple, stable, and efficient estimator for a flexible class of MNAR/pattern-mixture models, extending the pattern-graph framework from density/imputation settings to general estimating equations. The efficiency-bound calculation in Theorem 4.2 is nontrivial, and the bracketing-number lemmas (H.12-H.15) are potentially reusable. The simulations show improvements over entropy and local balancing. However, the central asymptotic proof currently has a load-bearing gap and an apparent mismatch with the algorithm actually proposed, so the efficiency claim is not yet supported.

major comments (3)

[Appendix F (proof of Theorem D.4)] The decomposition at the start of the proof of Theorem D.4 is not the decomposition for θ_seq as defined in Algorithm 1. The proof writes P̂_N ψ_θ − E{ψ_θ(L)} = Σ_{r∈R} [N^{-1}Σ_i 1_{R_i=1_d} O_r(L[r]_i; α̂[r]) ψ_θ(L_i) − E{1_{R_i=r} ψ_θ(L)}]. But Algorithm 1 defines weights as ŵ = Σ_r Q̂_r, with Q̂_r = Ô_r Σ_{s∈Pa(r)} Q̂_s, so the complete-case weighted sum contains products of odds along paths, not the single factors Ô_r. Unless O_r is redefined to mean Q_r, which the paper does not do, the proof establishes asymptotic normality for a different estimator, and the influence function displayed at the end of Appendix F is not the V_θ of Theorem 4.2 for a nontrivial pattern graph. This mismatch directly undermines Theorem 4.3 as stated.
[Appendix H, Lemma H.8] The proof of Lemma H.8 asserts the key balancing-error bound sup_θ |√N S^r_{θ,3}| ≤ λ{γ√K_r + 2(1−γ)√PEN2(Φ_r^T α̂[r])}√PEN2(Φ_r^T β^r_θ) and then concludes op(1). The inequality is never derived, and the notation PEN2 and γ is not defined anywhere in the paper. No argument shows that √PEN2(Φ_r^T β^r_θ) is uniformly bounded in a way that makes the product negligible, nor is the dependence on the recursive estimator Q̂_Pa(r) controlled. Since S^r_{θ,3} is precisely the balancing error that the sequential loss is designed to enforce, this is a load-bearing gap in the proof of Theorem D.4 and, consequently, in the efficiency claim of Theorem 4.3.
[Assumptions D.1.A and Theorem 4.3] The uniform-overlap condition c_0 ≤ O_r(l[r]) ≤ C_0 in Assumption D.1.A is used at several load-bearing points, including Theorem D.2, Lemmas H.4-H.5, and the proof of Theorem D.4, but it involves distributions of missing variables and cannot be verified from the observed data alone. The manuscript should state explicitly that the efficiency guarantee is conditional on this assumption and should provide a sensitivity analysis or a diagnostic for the overlap condition in the simulations; otherwise the practical scope of Theorem 4.3 is unclear.

minor comments (4)

[Section 5 (Simulation)] The text says that the local estimations 'fail in around 5% dataset' under the simulation setting; please define what constitutes failure, for example non-convergence, extreme estimated weights, or undefined estimates.
[Section 6 and Figure 5] Figure 5 labels patterns with six digits (e.g., 111111, 111101) while the analysis reports five predictors; please reconcile the dimensions of the patterns with the variables used.
[Algorithm 1] The 'Input' line describes {1_{R_i=1_d} Q̂_Pa(r)(L_i)} as input, but these quantities are produced by earlier recursive steps; please rephrase to indicate that they are available from previous stages of the algorithm.
[References] The reference to Dong, Wong, and Chan (2024) is listed as 'Balancing method for non-monotone missing data' without a publication venue; if this is a preprint or dissertation chapter, please indicate its status.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the efficiency bound is derived, and the proposed estimator is shown to attain it through a standard influence-function expansion.

full rationale

I walked the derivation chain from the identifying assumptions (pattern graphs, mixture coefficients, recursive odds), through the weighted estimating equations, the efficiency bound (Theorem 4.2), and the asymptotic expansion for the proposed estimator (Theorems D.4 and 4.3). The efficiency bound is derived from standard semiparametric pathwise-differentiability arguments citing Newey (1990) and Bickel et al. (1993), not assumed as an input. The influence function in the asymptotic normality proof is explicitly the variance of the weighted estimator's influence function, and the residual terms S^r_{theta,1}, S^r_{theta,2}, S^r_{theta,3} are shown to be o_p(N^{-1/2}) under the stated conditions, so the estimator attains the derived bound. The self-citations to Dong et al. (2024) and Chen (2022) are contextual: they describe prior balancing weights and pattern graphs, but the central consistency and efficiency argument is self-contained and does not reduce to those citations. The skeptical concern about Lemma H.8 is a possible proof gap because the balancing-error inequality is asserted rather than fully derived; however, that is a technical completeness issue, not circularity. The inequality is not equivalent to the conclusion by construction, and no fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' own prior work, and no ansatz is smuggled in solely via citation. Therefore no significant circularity is found.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The method introduces no new physical or probabilistic entities: pattern graphs, propensity odds, and balancing weights all come from prior work. The load-bearing inputs are modeling assumptions: the pattern graph identification structure, bounded propensity odds, a unique population minimizer, and sieve approximation rates. The only user-chosen tuning elements are the penalty λ, the basis functions, and in simulations the DGP family.

free parameters (3)

Penalty tuning parameter λ (with per-coordinate weights t_k) = chosen by cross-validation in simulations; theory requires λ = o(1/sqrt(K_r N_r))
The sequential balancing loss includes an L1 penalty whose strength and coordinate weights are user-selected. They affect finite-sample performance but vanish asymptotically under Assumption D.1.G.
Basis functions Φ_r(l[r]) for each pattern = user-specified; simulations use six splines of degree up to four per continuous variable and indicators for discrete…
The propensity odds are modeled as exp(Φ_r(l[r])^T α[r]); the choice of basis controls approximation error and is a modeling decision, although the theory allows the number of basis functions to grow with sample size.
Simulation DGP polynomial degree for true propensity odds = degrees up to four
The true missingness probabilities in Section 5 are generated from polynomials of degree up to four, which are well approximated by the splines of degree up to four used by the estimators, making the main simulation close to a correctly specified setting.

assumptions (6)

domain assumption Regular pattern graph identifying assumption: for each missing pattern r, p(l̄_r | l[r], R=r) = Σ_{s∈Pa(r)} C_{s,r}(l[r]) p(l̄_r | l[r], R=s).
This is the identification assumption that makes the full-data distribution recoverable; if the graph or mixture coefficients are misspecified, the efficient estimator targets the wrong parameter. Introduced in Sections 2.2 and 2.3.
domain assumption Overlap and boundedness: c0 ≤ O_r(l[r]) ≤ C0 for all r and l[r] (Assumption D.1.A).
Ensures inverse propensity weights are well behaved and gives sup-norm consistency of the odds estimators; equivalent to P(R=r | l[r], R∈{Pa(r),r}) bounded away from 0 and 1.
domain assumption Unique minimizer of the population sequential loss (Assumption D.1.B).
Identifiability of the odds model coefficients α0 is needed for consistency and cannot be verified from data.
ad hoc to paper Sieve approximation rates: O_r and u_rθ are approximable by basis functions at rates μ1, μ2 > 1/2 (Assumptions D.1.D and D.3.A).
These rate conditions ensure that approximation bias vanishes fast enough for the remainder terms in the asymptotic decomposition; they are standard in sieve estimation but are technical sufficient conditions rather than data-driven facts.
standard math Standard semiparametric efficiency and empirical process theory (Newey 1990; Bickel et al. 1993; Van der Vaart 2000).
Used to characterize the tangent space, compute the efficiency bound, and prove uniform convergence and Donsker properties in Appendices C and F-H.
domain assumption Bracketing entropy conditions on the propensity odds class M_r and estimating function class H (Assumptions D.3.B and D.3.C).
Controls the complexity of function classes so that empirical processes converge uniformly; satisfied by smooth function classes but assumed in the theorem.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Efficient Estimation under Multiple Missing Patterns via Balancing Weights." pith.science (2026). https://pith.science/paper/UYUCAFK2

@misc{pith2026250413467,
  author       = {Pith},
  title        = {Pith review of: Efficient Estimation under Multiple Missing Patterns via Balancing Weights},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UYUCAFK2}},
  note         = {Machine review of arXiv:2504.13467}
}

read the original abstract

As one of the most commonly seen data challenges, missing data, in particular, multiple, non-monotone missing patterns, complicates estimation and inference due to the fact that missingness mechanisms are often not missing at random, and conventional methods cannot be applied. Pattern graphs have recently been proposed as a tool to systematically relate various observed patterns in the sample. We extend its scope to the estimation of parameters defined by moment equations, including common regression models, via solving weighted estimating equations with weights constructed using a sequential balancing approach. These novel weights are carefully crafted to address the instability issue of the straightforward approach based on local balancing. We derive the efficiency bound for the model parameters and show that our proposed method, albeit relatively simple, is asymptotically efficient. Simulation results demonstrate the superior performance of the proposed method, and real-data applications illustrate how the results are robust to the choice of identification assumptions.

Figures

Figures reproduced from arXiv: 2504.13467 by the authors.

**Figure 1.** Examples of regular pattern graphs. Since a parent pattern is more informative than its child pattern, Chen (2022) models the unobserved part of pattern r using the information from its parents. Let Pa(r) be the set of parents of a pattern r ∈ R. Specifically, for any pattern r ∈ G and r 6= 1d, the identification assumption being encoded in G is p(l [r] | l [r] , R = r) = p(l [r] | l [r] , R ∈ Pa(r)) . (2) It can be… view at source ↗

**Figure 2.** Mixture coefficients encoded in regular pattern gra [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** A regular pattern graph for simulation We first analyzed the simulated data with the full dataset (Full), which is the ideal case with no missingness. We then analyzed the data in the complete case pattern (Completecase), for which data in all missing patterns r 6= 1d are discarded, and an unweighted analysis is used for the remaining data. Next, we considered the inverse propensity weighting methods with the true … view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A regular pattern graph for simulation the weighted estimating equation: N −1 PN i=1 1Ri=1dψθ(Li)/πˆ(Li) = 0 with corresponding loss. The biases and mean squared errors of each coefficient are shown in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: A regular pattern graph for real data analysis [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

28 extracted references · 22 canonical work pages

[1]

Begun, J. M., W. J. Hall, W.-M. Huang, and J. A. Wellner (1983). Information and asymptotic efficiency in parametric-nonparametric models. The Annals of Statistics\/ 11\/ (2), 432--452

work page 1983
[2]

Malinsky, and I

Bhattacharya, R., D. Malinsky, and I. Shpitser (2020). Causal inference under interference and network uncertainty. In Uncertainty in Artificial Intelligence , pp.\ 1028--1038. PMLR

work page 2020
[3]

Bickel, P. J., C. A. Klaassen, P. J. Bickel, Y. Ritov, J. Klaassen, J. A. Wellner, and Y. Ritov (1993). Efficient and adaptive estimation for semiparametric models , Volume 4. Springer

1993
[4]

Burns, W. J., E. Peters, and P. Slovic (2012). Risk perception and the economic crisis: A longitudinal study of the trajectory of perceived risk. Risk Analysis: An International Journal\/ 32\/ (4), 659--677

work page 2012
[5]

Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models. Handbook of econometrics\/ 6 , 5549--5632

2007
[6]

Hong, and A

Chen, X., H. Hong, and A. Tarozzi (2008). Semiparametric efficiency in gmm models with auxiliary data. The Annals of Statistics\/ 36\/ (2), 808--843

work page 2008
[7]

Chen, Y.-C. (2022). Pattern graphs: a graphical approach to nonmonotone missing data. The Annals of Statistics\/ 50\/ (1), 129--146

work page 2022
[8]

Dong, J., R. K. W. Wong, and K. C. G. Chan (2024). Balancing method for non-monotone missing data

work page 2024

Show all 28 references

[9]

Fan, J., K. Imai, I. Lee, H. Liu, Y. Ning, and X. Yang (2022). Optimal covariate balancing conditions in propensity score estimation. Journal of Business & Economic Statistics\/ 41\/ (1), 97--110

2022
[10]

Horowitz, J. L. and E. Mammen (2004). Nonparametric estimation of an additive model with a link function. The Annals of Statistics\/ 32\/ (6), 2412 -- 2443

2004
[11]

Ibragimov, I. A. and R. Z. Has' Minskii (2013). Statistical estimation: asymptotic theory , Volume 16. Springer Science & Business Media

2013
[12]

Little, R. J. (1993). Pattern-mixture models for multivariate incomplete data. Journal of the American Statistical Association\/ 88\/ (421), 125--134

1993
[13]

Mohan, K. and J. Pearl (2021). Graphical models for processing missing data. Journal of the American Statistical Association\/ 116\/ (534), 1023--1037

2021
[14]

Michiels, M

Molenberghs, G., B. Michiels, M. G. Kenward, and P. J. Diggle (1998). Monotone missing data and pattern-mixture models. Statistica Neerlandica\/ 52\/ (2), 153--161

1998
[15]

Bhattacharya, and I

Nabi, R., R. Bhattacharya, and I. Shpitser (2020). Full law identification in graphical models of missing data: Completeness results. In International conference on machine learning , pp.\ 7153--7163. PMLR

2020
[16]

Newey, W. K. (1990). Semiparametric efficiency bounds. Journal of applied econometrics\/ 5\/ (2), 99--135

1990
[17]

Newey, W. K. (1997). Convergence rates and asymptotic normality for series estimators. Journal of econometrics\/ 79\/ (1), 147--168

1997
[18]

Robins, J. M. (1997). Non-response models for the analysis of non-monotone non-ignorable missing data. Statistics in medicine\/ 16\/ (1), 21--37

1997
[19]

Rubin, D. B. (1976). Inference and missing data. Biometrika\/ 63\/ (3), 581--592

1976
[20]

Shpitser, I. (2016). Consistent estimation of functions of data missing non-monotonically and not at random. Advances in Neural Information Processing Systems\/ 29

2016
[21]

Tchetgen, E. J. T., L. Wang, and B. Sun (2018). Discrete choice models for nonmonotone nonignorable missing data: Identification and inference. Statistica Sinica\/ 28\/ (4), 2069

2018
[22]

Molenberghs, B

Thijs, H., G. Molenberghs, B. Michiels, G. Verbeke, and D. Curran (2002). Strategies to fit pattern-mixture models. Biostatistics\/ 3\/ (2), 245--265

2002
[23]

Troxel, A. B., D. P. Harrington, and S. R. Lipsitz (1998). Analysis of longitudinal data with non-ignorable non-monotone missing values. Journal of the Royal Statistical Society: Series C (Applied Statistics)\/ 47\/ (3), 425--438

1998
[24]

Van der Vaart, A. W. (2000). Asymptotic statistics , Volume 3. Cambridge university press

2000
[25]

Wellner, J. et al. (2013). Weak convergence and empirical processes: with applications to statistics . Springer Science & Business Media

2013
[26]

Wong, R. K. and K. C. G. Chan (2018). Kernel-based covariate functional balancing for observational studies. Biometrika\/ 105\/ (1), 199--213

2018
[27]

Zhao, Q. (2019). Covariate balancing propensity score by tailored loss functions. The Annals of Statistics\/ 47\/ (2), 965--993

2019
[28]

Zubizarreta, J. R. (2015). Stable weights that balance covariates for estimation with incomplete outcome data. Journal of the American Statistical Association\/ 110\/ (511), 910--922

2015

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Begun, J. M., W. J. Hall, W.-M. Huang, and J. A. Wellner (1983). Information and asymptotic efficiency in parametric-nonparametric models. The Annals of Statistics\/ 11\/ (2), 432--452

work page 1983

[2] [2]

Malinsky, and I

Bhattacharya, R., D. Malinsky, and I. Shpitser (2020). Causal inference under interference and network uncertainty. In Uncertainty in Artificial Intelligence , pp.\ 1028--1038. PMLR

work page 2020

[3] [3]

Bickel, P. J., C. A. Klaassen, P. J. Bickel, Y. Ritov, J. Klaassen, J. A. Wellner, and Y. Ritov (1993). Efficient and adaptive estimation for semiparametric models , Volume 4. Springer

1993

[4] [4]

Burns, W. J., E. Peters, and P. Slovic (2012). Risk perception and the economic crisis: A longitudinal study of the trajectory of perceived risk. Risk Analysis: An International Journal\/ 32\/ (4), 659--677

work page 2012

[5] [5]

Chen, X. (2007). Large sample sieve estimation of semi-nonparametric models. Handbook of econometrics\/ 6 , 5549--5632

2007

[6] [6]

Hong, and A

Chen, X., H. Hong, and A. Tarozzi (2008). Semiparametric efficiency in gmm models with auxiliary data. The Annals of Statistics\/ 36\/ (2), 808--843

work page 2008

[7] [7]

Chen, Y.-C. (2022). Pattern graphs: a graphical approach to nonmonotone missing data. The Annals of Statistics\/ 50\/ (1), 129--146

work page 2022

[8] [8]

Dong, J., R. K. W. Wong, and K. C. G. Chan (2024). Balancing method for non-monotone missing data

work page 2024