Robust and Semiparametric Methods in Causal Inference and Treatment Effect Estimation

Unmeasured confounders can introduce significant bias in causal inference, leading to incorrect conclusions. To address this issue, we propose a novel calibration method leveraging multiple candidate negative control outcomes (NCOs), including potentially invalid ones. Unlike traditional proximal inference methods, our distributional NCO framework assumes that system bias arises from an underlying distribution, rather than treating it as a fixed parameter. We develop a semiparametric approach that effectively removes the influence of this bias distribution, enabling more robust statistical inference on causal effects. Our method provides a flexible way to correct for confounding while accommodating uncertainty in NCO validity. We evaluate its performance through simulations and a real-world application. The results highlight the effectiveness of our framework in improving causal effect estimation under unmeasured confounding. Our approach extends the applicability of negative control methods, offering a more generalizable solution for bias correction in observational studies. 

This project aims to extend Difference-in-Differences (DiD) methods in a potential outcome framework to study causal relationships for random object responses and continuous treatments in the presence of high-dimensional confounders, with a focus on large-scale observational studies. Motivated by assessing the causal link between air pollution and health outcomes going beyond traditional regression methods. We appropriately define the causal effects for varying levels of the continuous treatment by utilizing a Hilbert space embedding of the metric space valued outcomes, propose a doubly debiased estimator via data splitting, and analyze its asymptotic properties. 

Weak identification is a common issue for many statistical problems. In instrumental variable literature, weak instruments problem arises when the instruments are weakly correlated with treatment; in proximal causal inference literature, weak proxies issue can also arise when the proxies are weakly correlated with unmeasured confounders. Under weak-identification, standard estimation methods, such as Generalized method of moments (GMM), have poor finite sample performance. In this paper, we studied the estimation and inference problem with many weak moment conditions with Neyman orthogonality. We developed a two-step continuous updating estimator which allows first step possibly non-parametric estimation for nuisance parameter, and proved its consistency and asymptotic normality. Our theory is applicable even when there are high-dimensional moment conditions and high-dimensional nuisance parameters. We applied our general theory to study estimation and inference for weak instruments and weak proxies problems. 

Regression discontinuity (RD) designs exploit locally random sorting across a treatment threshold in a running variable to estimate intervention effects. When the running variable is temporal (e.g., time or age), treatment is assigned at a time threshold, and all units eventually receive it, distinguishing Temporal Regression Discontinuity (TRD) from cross-sectional RD. While constant post-treatment effects are often assumed, complex processes in fields like environmental science and public health make this unrealistic. Instead, treatment effects evolve over time. To address this, we propose a flexible Gaussian process-based approach to detect and quantify time-varying effects with few assumptions. Our framework introduces an effect function to capture post-treatment dynamics and provides conditions to extract effect magnitude and duration via its derivative. We also develop a robust statistical test for time-varying effects. Simulations demonstrate our method's superiority over OLS and local regression in TRD settings. Finally, we apply our method to evaluate the impacts of a major transport intervention on air quality. 

This paper evaluates how small perturbations in covariate distributions affect Quantile Treatment Effect (QTE). We introduce a new metric to quantify the sensitivity of QTE to such shifts across various quantile levels. We propose corresponding point and variance estimators and establish the asymptotic properties. The performance of our method is supported by simulation studies. Additionally, we apply our approach to the 2017-2018 National Health and Nutrition Examination Survey (NHANES) data, and find that counterfactually increasing the proportion of females in the population significantly reduces the QTE at lower percentiles, indicating that maintaining a sufficient vitamin D level is particularly effective in increasing bone mineral density for males with lower bone mineral density. 

Targeted Maximum Likelihood Estimation (TMLE) is a plug-in estimator for a target estimand that is asymptotically efficient under regularity conditions. A key component of TMLE is the targeting step, which maps an initial estimator $P_n^0$ of the true data distribution $P_0$ to an updated estimator $P_n^*$ that solves the score equation $P_nD(P_n^*) \approx 0$, where $D$ denotes the canonical gradient (also called the efficient influence curve, EIC). The targeting step is specified by choosing a least-favorable model (LFM) $\{P_t\}$ with an offset $P$--for example, the initial estimator of the data distribution--and performing iterative maximum-likelihood updates along this path until the EIC score equation is satisfied. A local LFM is a parametric model that spans a score at the offset ($t=0$) equal to the canonical gradient of the pathwise derivative of the target estimand. A local LFM often exists in closed form but is not unique, and it generally requires iterative MLE updates to ensure that the EIC score equation is solved to the desired precision.

Van der Laan \& Gruber (2016) introduced the universal least favorable model (ULFM), defined as the integral curve of the vector field $\partial_t\log p_t(x)=D(P_t)(x)$, whose score remains proportional to the EIC for all $t$. They observed that a single MLE update along this model exactly solves the EIC score equation, hence makes iterative targeting unnecessary, and makes this one-step TMLE a statistically robust choice. However, because explicit ULFMs are rarely available, the one-step TMLE can be computationally demanding--especially when evaluating the canonical gradient at a given measure $P$ is challenging: the one-step approach may require many such evaluations, whereas iterative TMLE typically requires only a few. Thus, there is a trade-off between the statistical performance of the ULFM-based one-step TMLE and the computational simplicity of the iterative TMLE based on an explicit local LFM. To navigate this trade-off, we construct a hierarchy of $k$th-order local paths whose Taylor expansions match that of the ULFM up through order $k$. A single targeting step along such a path cancels successive terms in the expansion of the first-order bias and yields $P_nD(P_n^*)=o_p(\|P_n^0-P_0\|^{k+1})$, and thereby sharpening the convergence rate of the efficient score compared to a TMLE based on a standard local LFM.

One contribution here is to supply--under mild $L^{2}$ continuity of the map $P\mapsto D(P)$--general existence and uniqueness results by formulating the ULFM PDE as an ODE in a Banach space, and to supply an explicit ODE representation for the $k$th-order local paths that directly underpins their numerical implementation. Pointwise error estimates show that these truncated paths approximate the ULFM to order $k+1$ for sufficiently small $t$. We illustrate this construction with closed-form examples for population moments, smooth density functionals, and the average treatment effect. Empirical studies with the quadratic functional $\Psi(P)=\int p^2(x) dx$ show that replacing a first-order local path with its second-order approximation reduces the EIC score error by at least a factor of the convergence rate of the initial estimator, producing noticeable finite-sample gains when the true density is sharply curved, even though the ultimate convergence rate remains governed by the von Mises remainder.

Finally, we analyze the limitations of $k$th-order approximations when the efficient influence curve $D(P)$ lacks smoothness in $P$--for example, in estimating the median--showing that our Banach-space existence and error bounds can break down for the truncated ODE. To address this, we introduce a simple smoothing of the parameter map that restores the mild $L^2$ continuity required for well-posedness of the $k$th-order local path. These results provide an explicit, implementable ODE formulation for one-step TMLE updates using higher-order local approximations of the ULFM, and make transparent exactly how the truncation order controls the first-order bias term in semiparametric inference.