In experimental and observational studies, there is often interest in understanding the mechanism through which an intervention program improves the final outcome. Causal mediation analyses have been developed for this purpose but are primarily considered for the case of perfect treatment compliance, with a few exceptions that require the exclusion restriction assumption. In this article, we consider a semiparametric framework for assessing causal mediation in the presence of treatment noncompliance without the exclusion restriction. We propose a set of assumptions to identify the natural mediation effects for the entire study population and further, for the principal natural mediation effects within subpopulations characterized by the potential compliance behavior. We derive the efficient influence functions for the principal natural mediation effect estimands and motivate a set of multiply robust estimators for inference. The multiply robust estimators remain consistent to their respective estimands under four types of misspecification of the working models and are efficient when all nuisance models are correctly specified. We further introduce a nonparametric extension of the proposed estimators by incorporating machine learners to estimate the nuisance functions. Sensitivity analysis methods are also discussed for addressing key identification assumptions. We demonstrate the proposed methods via simulations and an application to a real data example.
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We observe n possibly dependent random variables, the distribution of which is presumed to be stationary even though this might not be true, and we aim at estimating the stationary distribution. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and our estimator. If the dependence within the observations is small, the estimator performs as good as if the data were independent and identically distributed. In addition our estimator is robust to misspecification and contamination. If the dependence is too high but the observed process is mixing, we can select a subset of observations that is almost independent and retrieve results similar to what we have in the i.i.d. case. We apply our procedure to the estimation of the invariant distribution of a diffusion process and to finite state space hidden Markov models.
Forward simulation-based uncertainty quantification that studies the distribution of quantities of interest (QoI) is a crucial component for computationally robust engineering design and prediction. There is a large body of literature devoted to accurately assessing statistics of QoIs, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems.
Mediation analysis in causal inference typically concentrates on one binary exposure, using deterministic interventions to split the average treatment effect into direct and indirect effects through a single mediator. Yet, real-world exposure scenarios often involve multiple continuous exposures impacting health outcomes through varied mediation pathways, which remain unknown a priori. Addressing this complexity, we introduce NOVAPathways, a methodological framework that identifies exposure-mediation pathways and yields unbiased estimates of direct and indirect effects when intervening on these pathways. By pairing data-adaptive target parameters with stochastic interventions, we offer a semi-parametric approach for estimating causal effects in the context of high-dimensional, continuous, binary, and categorical exposures and mediators. In our proposed cross-validation procedure, we apply sequential semi-parametric regressions to a parameter-generating fold of the data, discovering exposure-mediation pathways. We then use stochastic interventions on these pathways in an estimation fold of the data to construct efficient estimators of natural direct and indirect effects using flexible machine learning techniques. Our estimator proves to be asymptotically linear under conditions necessitating n to the negative quarter consistency of nuisance function estimation. Simulation studies demonstrate the square root n consistency of our estimator when the exposure is quantized, whereas for truly continuous data, approximations in numerical integration prevent square root n consistency. Our NOVAPathways framework, part of the open-source SuperNOVA package in R, makes our proposed methodology for high-dimensional mediation analysis available to researchers, paving the way for the application of modified exposure policies which can delivery more informative statistical results for public policy.
Observational studies are frequently used to estimate the effect of an exposure or treatment on an outcome. To obtain an unbiased estimate of the treatment effect, it is crucial to measure the exposure accurately. A common type of exposure misclassification is recall bias, which occurs in retrospective cohort studies when study subjects may inaccurately recall their past exposure. Specifically, differential recall bias can be problematic when examining the effect of a self-reported binary exposure since the magnitude of recall bias can differ between groups. In this paper, we provide the following contributions: 1) we derive bounds for the average treatment effect (ATE) in the presence of recall bias; 2) we develop several estimation approaches under different identification strategies; 3) we conduct simulation studies to evaluate their performance under several scenarios of model misspecification; 4) we propose a sensitivity analysis method that can examine the robustness of our results with respect to different assumptions; and 5) we apply the proposed framework to an observational study, estimating the effect of childhood physical abuse on adulthood mental health.
We consider estimation and inference for a regression coefficient in panels with interactive fixed effects (i.e., with a factor structure). We show that previously developed estimators and confidence intervals (CIs) might be heavily biased and size-distorted when some of the factors are weak. We propose estimators with improved rates of convergence and bias-aware CIs that are uniformly valid regardless of whether the factors are strong or not. Our approach applies the theory of minimax linear estimation to form a debiased estimate using a nuclear norm bound on the error of an initial estimate of the interactive fixed effects. We use the obtained estimate to construct a bias-aware CI taking into account the remaining bias due to weak factors. In Monte Carlo experiments, we find a substantial improvement over conventional approaches when factors are weak, with little cost to estimation error when factors are strong.
In this paper, we consider estimation of average treatment effect on the treated (ATT), an interpretable and relevant causal estimand to policy makers when treatment assignment is endogenous. By considering shadow variables that are unrelated to the treatment assignment but related to interested outcomes, we establish identification of the ATT. Then we focus on efficient estimation of the ATT by characterizing the geometric structure of the likelihood, deriving the semiparametric efficiency bound for ATT estimation and proposing an estimator that can achieve this bound. We rigorously establish the theoretical results of the proposed estimator. The finite sample performance of the proposed estimator is studied through comprehensive simulation studies as well as an application to our motivating study.
Probability density estimation is a core problem of statistics and signal processing. Moment methods are an important means of density estimation, but they are generally strongly dependent on the choice of feasible functions, which severely affects the performance. In this paper, we propose a non-classical parametrization for density estimation using sample moments, which does not require the choice of such functions. The parametrization is induced by the squared Hellinger distance, and the solution of it, which is proved to exist and be unique subject to a simple prior that does not depend on data, and can be obtained by convex optimization. Statistical properties of the density estimator, together with an asymptotic error upper bound are proposed for the estimator by power moments. Applications of the proposed density estimator in signal processing tasks are given. Simulation results validate the performance of the estimator by a comparison to several prevailing methods. To the best of our knowledge, the proposed estimator is the first one in the literature for which the power moments up to an arbitrary even order exactly match the sample moments, while the true density is not assumed to fall within specific function classes.
Background: Policy evaluation studies that assess how state-level policies affect health-related outcomes are foundational to health and social policy research. The relative ability of newer analytic methods to address confounding, a key source of bias in observational studies, has not been closely examined. Methods: We conducted a simulation study to examine how differing magnitudes of confounding affected the performance of four methods used for policy evaluations: (1) the two-way fixed effects (TWFE) difference-in-differences (DID) model; (2) a one-period lagged autoregressive (AR) model; (3) augmented synthetic control method (ASCM); and (4) the doubly robust DID approach with multiple time periods from Callaway-Sant'Anna (CSA). We simulated our data to have staggered policy adoption and multiple confounding scenarios (i.e., varying the magnitude and nature of confounding relationships). Results: Bias increased for each method: (1) as confounding magnitude increases; (2) when confounding is generated with respect to prior outcome trends (rather than levels), and (3) when confounding associations are nonlinear (rather than linear). The AR and ASCM have notably lower root mean squared error than the TWFE model and CSA approach for all scenarios; the exception is nonlinear confounding by prior trends, where CSA excels. Coverage rates are unreasonably high for ASCM (e.g., 100%), reflecting large model-based standard errors and wide confidence intervals in practice. Conclusions: Our simulation study indicated that no single method consistently outperforms the others. But a researcher's toolkit should include all methodological options. Our simulations and associated R package can help researchers choose the most appropriate approach for their data.
In this paper, we consider estimating spot/instantaneous volatility matrices of high-frequency data collected for a large number of assets. We first combine classic nonparametric kernel-based smoothing with a generalised shrinkage technique in the matrix estimation for noise-free data under a uniform sparsity assumption, a natural extension of the approximate sparsity commonly used in the literature. The uniform consistency property is derived for the proposed spot volatility matrix estimator with convergence rates comparable to the optimal minimax one. For the high-frequency data contaminated by microstructure noise, we introduce a localised pre-averaging estimation method that reduces the effective magnitude of the noise. We then use the estimation tool developed in the noise-free scenario, and derive the uniform convergence rates for the developed spot volatility matrix estimator. We further combine the kernel smoothing with the shrinkage technique to estimate the time-varying volatility matrix of the high-dimensional noise vector. In addition, we consider large spot volatility matrix estimation in time-varying factor models with observable risk factors and derive the uniform convergence property. We provide numerical studies including simulation and empirical application to examine the performance of the proposed estimation methods in finite samples.
This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.
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