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1. A General Framework for Treatment Effect Estimation in Semi-Supervised and High Dimensional Settings

   Chakrabortty, Abhishek; Dai, Guorong (August 2024, arXiv.org)

   In this article, we aim to provide a general and complete understanding of semi-supervised (SS) causal inference for treatment effects. Specifically, we consider two such estimands: (a) the average treatment effect and (b) the quantile treatment effect, as prototype cases, in an SS setting, characterized by two available data sets: (i) a labeled data set of size n, providing observations for a response and a set of high dimensional covariates, as well as a binary treatment indicator; and (ii) an unlabeled data set of size N, much larger than n, but without the response observed. Using these two data sets, we develop a family of SS estimators which are ensured to be: (1) more robust and (2) more efficient than their supervised counterparts based on the labeled data set only. Beyond the 'standard' double robustness results (in terms of consistency) that can be achieved by supervised methods as well, we further establish root-n consistency and asymptotic normality of our SS estimators whenever the propensity score in the model is correctly specified, without requiring specific forms of the nuisance functions involved. Such an improvement of robustness arises from the use of the massive unlabeled data, so it is generally not attainable in a purely supervised setting. In addition, our estimators are shown to be semi-parametrically efficient as long as all the nuisance functions are correctly specified. Moreover, as an illustration of the nuisance estimators, we consider inverse-probability-weighting type kernel smoothing estimators involving unknown covariate transformation mechanisms, and establish in high dimensional scenarios novel results on their uniform convergence rates, which should be of independent interest. Numerical results on both simulated and real data validate the advantage of our methods over their supervised counterparts with respect to both robustness and efficiency. 

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2. The Decaying Missing-at-Random Framework: Model Doubly Robust Causal Inference with Partially Labeled Data

   Zhang, Yuqian; Chakrabortty, Abhishek; Bradic, Jelena (April 2025, arXiv.org)

   In modern large-scale observational studies, data collection constraints often result in partially labeled datasets, posing challenges for reliable causal inference, especially due to potential labeling bias and relatively small size of the labeled data. This paper introduces a decaying missing-at-random (decaying MAR) framework and associated approaches for doubly robust causal inference on treatment effects in such semi-supervised (SS) settings. This simultaneously addresses selection bias in the labeling mechanism and the extreme imbalance between labeled and unlabeled groups, bridging the gap between the standard SS and missing data literatures, while throughout allowing for confounded treatment assignment and high-dimensional confounders under appropriate sparsity conditions. To ensure robust causal conclusions, we propose a bias-reduced SS (BRSS) estimator for the average treatment effect, a type of 'model doubly robust' estimator appropriate for such settings, establishing asymptotic normality at the appropriate rate under decaying labeling propensity scores, provided that at least one nuisance model is correctly specified. Our approach also relaxes sparsity conditions beyond those required in existing methods, including standard supervised approaches. Recognizing the asymmetry between labeling and treatment mechanisms, we further introduce a de-coupled BRSS (DC-BRSS) estimator, which integrates inverse probability weighting (IPW) with bias-reducing techniques in nuisance estimation. This refinement further weakens model specification and sparsity requirements. Numerical experiments confirm the effectiveness and adaptability of our estimators in addressing labeling bias and model misspecification. 

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3. Bayesian semi-supervised inference via a debiased modeling approach

   https://doi.org/10.1016/j.ecosta.2025.05.001  

   Sert, Gözde; Chakrabortty, Abhishek; Bhattacharya, Anirban (May 2025, Econometrics and Statistics)

   Inference in semi-supervised (SS) settings has gained substantial attention in recent years due to increased relevance in modern big-data problems. In a typical SS setting, there is a much larger-sized unlabeled data, containing only observations of predictors, and a moderately sized labeled data containing observations for both an outcome and the set of predictors. Such data naturally arises when the outcome, unlike the predictors, is costly or difficult to obtain. One of the primary statistical objectives in SS settings is to explore whether parameter estimation can be improved by exploiting the unlabeled data. We propose a novel Bayesian method for estimating the population mean in SS settings. The approach yields estimators that are both efficient and optimal for estimation and inference. The method itself has several interesting artifacts. The central idea behind the method is to model certain summary statistics of the data in a targeted manner, rather than the entire raw data itself, along with a novel Bayesian notion of debiasing. Specifying appropriate summary statistics crucially relies on a debiased representation of the population mean that incorporates unlabeled data through a flexible nuisance function while also learning its estimation bias. Combined with careful usage of sample splitting, this debiasing approach mitigates the effect of bias due to slow rates or misspecification of the nuisance parameter from the posterior of the final parameter of interest, ensuring its robustness and efficiency. Concrete theoretical results, via Bernstein--von Mises theorems, are established, validating all claims, and are further supported through extensive numerical studies. To our knowledge, this is possibly the first work on Bayesian inference in SS settings, and its central ideas also apply more broadly to other Bayesian semi-parametric inference problems. 

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4. Optimal subsampling for quantile regression in big data

   https://doi.org/10.1093/biomet/asaa043  

   Wang, Haiying; Ma, Yanyuan (July 2020, Biometrika)

   null (Ed.)

   Summary We investigate optimal subsampling for quantile regression. We derive the asymptotic distribution of a general subsampling estimator and then derive two versions of optimal subsampling probabilities. One version minimizes the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator and the other minimizes that of the original parameter estimator. The former does not depend on the densities of the responses given covariates and is easy to implement. Algorithms based on optimal subsampling probabilities are proposed and asymptotic distributions, and the asymptotic optimality of the resulting estimators are established. Furthermore, we propose an iterative subsampling procedure based on the optimal subsampling probabilities in the linearly transformed parameter estimation which has great scalability to utilize available computational resources. In addition, this procedure yields standard errors for parameter estimators without estimating the densities of the responses given the covariates. We provide numerical examples based on both simulated and real data to illustrate the proposed method. 

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5. Robust estimation and inference for expected shortfall regression with many regressors

   https://doi.org/10.1093/jrsssb/qkad063  

   He, Xuming; Tan, Kean_Ming; Zhou, Wen-Xin (June 2023, Journal of the Royal Statistical Society Series B: Statistical Methodology)

   Abstract Expected shortfall (ES), also known as superquantile or conditional value-at-risk, is an important measure in risk analysis and stochastic optimisation and has applications beyond these fields. In finance, it refers to the conditional expected return of an asset given that the return is below some quantile of its distribution. In this paper, we consider a joint regression framework recently proposed to model the quantile and ES of a response variable simultaneously, given a set of covariates. The current state-of-the-art approach to this problem involves minimising a non-differentiable and non-convex joint loss function, which poses numerical challenges and limits its applicability to large-scale data. Motivated by the idea of using Neyman-orthogonal scores to reduce sensitivity to nuisance parameters, we propose a statistically robust and computationally efficient two-step procedure for fitting joint quantile and ES regression models that can handle highly skewed and heavy-tailed data. We establish explicit non-asymptotic bounds on estimation and Gaussian approximation errors that lay the foundation for statistical inference, even with increasing covariate dimensions. Finally, through numerical experiments and two data applications, we demonstrate that our approach well balances robustness, statistical, and numerical efficiencies for expected shortfall regression. 

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