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Generalized entropy calibration for inference with partially observed data: A unified framework

Mst Moushumi Pervin, Hengfang Wang, Jae Kwang Kim

Abstract

Missing data is an universal problem in statistics. We develop a unified framework for estimating parameters defined by general estimating equations under a missing-at-random (MAR) mechanism, based on generalized entropy calibration weighting. We construct weights by minimizing a convex entropy subject to (i) balancing constraints on a data-adaptive calibration function, estimated using flexible machine-learning predictors with cross-fitting, and (ii) a debiasing constraint involving the fitted propensity score (PS) model. The resulting estimator is doubly robust, remaining consistent if either the outcome regression (OR) or the PS model is correctly specified, and attains the semiparametric efficiency bound when both models are correctly specified. Our formulation encompasses classical inverse probability weighting (IPW) and augmented IPW (AIPW) as special cases and accommodates a broad class of entropy functions. We illustrate the versatility of the approach in three important settings: semi-supervised learning with unlabeled outcomes, regression analysis with missing covariates, and causal effect estimation in observational studies. Extensive simulation studies and real-data applications demonstrate that the proposed estimators achieve greater efficiency and numerical stability than existing methods. In particular, the proposed estimator outperforms the classical AIPW estimator under the OR model misspecification.

Generalized entropy calibration for inference with partially observed data: A unified framework

Abstract

Missing data is an universal problem in statistics. We develop a unified framework for estimating parameters defined by general estimating equations under a missing-at-random (MAR) mechanism, based on generalized entropy calibration weighting. We construct weights by minimizing a convex entropy subject to (i) balancing constraints on a data-adaptive calibration function, estimated using flexible machine-learning predictors with cross-fitting, and (ii) a debiasing constraint involving the fitted propensity score (PS) model. The resulting estimator is doubly robust, remaining consistent if either the outcome regression (OR) or the PS model is correctly specified, and attains the semiparametric efficiency bound when both models are correctly specified. Our formulation encompasses classical inverse probability weighting (IPW) and augmented IPW (AIPW) as special cases and accommodates a broad class of entropy functions. We illustrate the versatility of the approach in three important settings: semi-supervised learning with unlabeled outcomes, regression analysis with missing covariates, and causal effect estimation in observational studies. Extensive simulation studies and real-data applications demonstrate that the proposed estimators achieve greater efficiency and numerical stability than existing methods. In particular, the proposed estimator outperforms the classical AIPW estimator under the OR model misspecification.
Paper Structure (53 sections, 10 theorems, 93 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 53 sections, 10 theorems, 93 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Existing Methods
  3. Proposed Estimators
  4. Dual formulation
  5. Large-Sample Properties
  6. Linearization
  7. Consistency under the correct PS model
  8. Consistency under the correct OR model
  9. Double robustness, variance dominance, and local efficiency
  10. Computational Details
  11. Profile optimization
  12. Calibration via cross-fitting
  13. Illustrative Examples
  14. Causal inference
  15. Semi-supervised learning
  16. ...and 38 more sections

Key Result

Theorem 4.1

Under Assumptions 1--4, where $s_i(\hat{\varphi}) = \bigl(b_i^\top,\, g(\pi^{-1}(O_i;\hat{\varphi}))\bigr)^\top$, and $\gamma^* \in \mathbb{R}^{q \times (q+1)}$ is the probability limit of $\hat{\gamma}$ satisfying with $f' = (g^{-1})'$. This expansion holds without assuming correctness of either the PS or the OR model.

Figures (5)

  • Figure 1: Estimation of Average Treatment Effects (ATE) under four scenarios: under OR1(2), the outcome regression (OR) model is correctly specified (misspecified); under PS1(2), the propensity score model is correctly specified (misspecified). The horizontal red line represents the true ATE.
  • Figure 2: Boxplots of the estimated linear regression coefficients $\bm{\beta}$ under OR2 (OR model misspecification) with MAR missingness.
  • Figure 3: Boxplots of the estimated linear regression coefficients $\bm{\beta}$ under OR2 (OR model misspecification) with MCAR missingness.
  • Figure 4: Estimation of the linear regression coefficients $\bm{\beta}$ under OR1 (when the OR model is correctly specified) and MCAR missingness mechanism.
  • Figure 5: Weighted covariate distributions for the lalonde1986evaluating data.

Theorems & Definitions (22)

  • Theorem 4.1: Linearization
  • Lemma 4.1
  • Corollary 1: Asymptotic normality under correct PS model
  • Lemma 4.2
  • Corollary 2: Asymptotic normality under correct OR model
  • Corollary 3: Variance dominance over AIPW
  • Remark 1
  • Remark 2
  • Lemma B.1: Consistency
  • proof
  • ...and 12 more