Experiment Design

Always Valid Inference: Bringing Sequential Analysis to A/B Testing

Ramesh Johari, Leo Pekelis, David J. Walsh · 2015 · 101 citations

A/B tests are typically analyzed via frequentist p-values and confidence intervals; but these inferences are wholly unreliable if users endogenously choose samples sizes by *continuously monitoring* their tests. We define *always valid* p-values and confidence intervals that let users try to take advantage of data as fast as it becomes available, providing valid statistical inference whenever they make their decision. Always valid inference can be interpreted as a natural interface for a sequential hypothesis test, which empowers users to implement a modified test tailored to them. In particular, we show in an appropriate sense that the measures we develop tradeoff sample size and power efficiently, despite a lack of prior knowledge of the user's relative preference between these two goals. We also use always valid p-values to obtain multiple hypothesis testing control in the sequential context. Our methodology has been implemented in a large scale commercial A/B testing platform to analyze hundreds of thousands of experiments to date.

Online controlled experiments at large scale

Ron Kohavi, Alex Deng, Brian Frasca +3 more · Knowledge Discovery and Data Mining · 2013 · 420 citations

Designing Persuasive Experiments

Karun Adusumilli, Abhi Vemulapati · 2026

Incentives in experimental design are often misaligned: experimenters design and finance experiments to seek regulatory approval, while regulators seek to maximize social-welfare. We propose a framework to resolve this conflict, wherein regulators set a minimum expected welfare threshold, and experimenters optimize designs subject to this constraint. It requires no knowledge of experimenters' private preferences or costs and mitigates strategic Bayesian persuasion. Under normal priors, sampling according to the Neyman-allocation is always optimal, independent of the specific objectives. Furthermore, we characterize the optimal stopping-rule. In a numerical study calibrated to historical clinical-trial data, our framework reduces expected sample-sizes by over 48% relative to classical designs that attain the same social-welfare.

TCARD: Nearly Balanced Two-Level Designs with Treatment Cardinality Constraints with an Application to LLM Prompt Engineering

Kexin Xie, Ryan Lekivetz, Xinwei Deng · 2026

Modern experimental designs often face the so-called treatment cardinality constraint, which is the constraint on the number of included factors in each treatment. Experiments with such constraints are commonly encountered in engineering simulation, AI system tuning, and large-scale system verification. This calls for the development of adequate designs to enable statistical efficiency for modeling and analysis within feasible constraints. In this work, we study two-level designs under this $k$-treatment cardinality constraint (TCARD), where the design matrix $\mathbf{X} \in \{0,1\}^{n \times p}$ has constant row sums equal to $k$. Although TCARDs are closely related to balanced incomplete block designs (BIBDs), exact BIBD structure is unavailable for many practical $(n,p,k)$ combinations. This leads to the notion of nearly balanced TCARDs, which we prove minimize the first two components of the generalized word-length pattern. We also show that good projection behavior in this setting is governed by two count-based regularities: balanced factor replications and uniform pairwise concurrences. Motivated by this characterization, we then propose the Balanced Concurrence Deviation ($Φ_{\mathrm{BCD}}$), a model-free objective that jointly penalizes replication imbalance and concurrence dispersion. We further show that this criterion is closely connected to classical optimality principles, including $(M,S)$-optimality, centered $\mathrm{UE}(s^2)$ criterion, and Bayesian $D$-optimality. To construct designs minimizing $Φ_{\mathrm{BCD}}$, we develop a coordinate-exchange (CE) algorithm with efficient incremental updates, together with a simulation-based procedure for calibrating the criterion weights to the intended downstream task. Numerical experiments confirm that the proposed method compares favorably with existing alternatives across a range of problem sizes and constraint strengths.

Mind the Sim-to-Real Gap & Think Like a Scientist

Harsh Parikh, Gabriel Levin-Konigsberg, Dominique Perrault-Joncas +1 more · 2026

Suppose a planner has a pre-trained simulator of a sequential decision problem and the option to run real experiments in the field. The simulator is cheap to query but inherits confounding and drift from its calibration data. Experimentation is unbiased but consumes one real unit per trial. We study when, and how, the planner should supplement the simulator with experiments. We give three results. First, an extended simulation lemma decomposes the simulator's value error into a calibration--deployment shift that randomization can identify and a parametric residual that no further interaction can reduce. Second, the value gap between the simulator-optimal policy and the optimum splits into a local component, on states the deployed policy already visits, and a reachability component, on states it does not. The reachability component stays bounded away from zero at any horizon under purely passive learning. Third, we propose Fisher-SEP, a simulation-aided experimental policy (SEP) that minimizes the posterior predictive variance of a target policy's value, with reward-only and transition-only specializations. Two case studies illustrate the regimes. In a vending-machine supply chain, front-loaded experimentation overtakes posterior updating once the horizon is long enough to amortize the pilot. In an HIV mobile-testing example with a corridor that separates a well-surveilled region from a poorly-surveilled one, only designed exploration reaches the poorly-surveilled region.

Valuing Winners: When and How to Correct for Selection Bias in Randomized Experiments

Ron Berman, Walter W. Zhang, Hangcheng Zhao · 2026

Decision-makers often deploy the best-performing treatment from a randomized experiment, creating a winner's curse: selection favors treatments whose observed outcomes are high partly because of statistical noise, so the naïve estimate of the winner is upward biased. We distinguish two forms of winner's curse, bias relative to the true best treatment (global) and bias relative to the selected treatment's true mean (selective), and link them to regret from deploying a suboptimal treatment. This framework defines seven decision-relevant evaluation targets: mean bias, mean squared error, and confidence interval coverage for the global and selective winner's curse, and mean regret. We then show that methods that perform well on one target can perform poorly on others, so corrections should be matched to the manager's objective. Across simulations with varying effect sizes, multiple-arm settings, and data calibrated to an online A/B testing platform, no method dominates uniformly: the plug-in estimator performs best when treatment differences are large, cross-fitting performs best when treatments are similar, and resampling methods often achieve low mean squared error for moderate differences. We also introduce an adaptive empirical likelihood procedure that delivers asymptotically valid confidence intervals across settings without the tuning sensitivity of resampling-based methods.

Minimax unbiased estimation for finite populations with bounded outcomes

P. M. Aronow, Patrick Lopatto · 2026

We study design-unbiased estimation of the finite-population total $\sum_{i=1}^N y_i$ when each outcome satisfies known bounds $y_i\in[a_i,b_i]$. For any sampling design with inclusion probabilities $π_i>0$, we prove a sharp lower bound on the worst-case squared error over the rectangular parameter space. This bound is attained if and only if the unit inclusion indicators are pairwise independent, in which case the minimax estimator is the midpoint-differenced Horvitz-Thompson estimator $\sum_{i=1}^N m_i+\sum_{i\in S}(y_i-m_i)/π_i$, with $m_i=(a_i+b_i)/{2}$. We then solve the joint design-and-estimation problem under the constraint $\sum_i π_i\le n$. We find that a minimax strategy samples units independently with probabilities $π_i^\ast=\min(1,c (b_i-a_i))$ where $c>0$ is chosen so that $\sum_i π_i^\ast=n$, and uses the midpoint-differenced estimator. This extends Gabler (1990)'s linear minimax result to the full class of design-unbiased estimators. We also show that the estimator is admissible among unbiased estimators and affine equivariant.

Sensitivity analysis for contamination in egocentric-network randomized trials with interference

Bar Weinstein, Daniel Nevo · 2026

Egocentric-Network Randomized Trials (ENRTs) are increasingly used to estimate causal effects under interference when measuring complete sociocentric network data is infeasible. ENRTs rely on egocentric network sampling, where a set of egos is first sampled, and each ego recruits a subset of its neighbors as alters. Treatments are then randomized across egos. While the observed ego-networks are disjoint by design, the underlying population network may contain edges connecting them, leading to contamination. Under a design-based framework, we show that the Horvitz-Thompson estimators of direct and indirect effects are biased whenever contamination is present. To address this, we derive bias-corrected estimators and propose a novel sensitivity analysis framework based on sensitivity parameters representing the probability or expected number of missing edges. This framework is implemented via both grid sensitivity analysis and probabilistic bias analysis, providing researchers with a flexible tool to assess the robustness of the causal estimators to contamination. We apply our methodology to the HIV Prevention Trials Network 037 study, finding that ignoring contamination may lead to underestimation of indirect effects and overestimation of direct effects.

Optimal Design under Interference, Homophily, and Robustness Trade-offs

Vydhourie Thiyageswaran, Alex Kokot, Jennifer Brennan +3 more · 2026

To minimize the mean squared error (MSE) in global average treatment effect (GATE) estimation under network interference, a popular approach is to use a cluster-randomized design. However, in the presence of homophily, which is common in social networks, cluster randomization can instead increase the MSE. We develop a novel potential outcomes model that accounts for interference, homophily, and heterogeneous variation. In this setting, we establish a framework for optimizing designs for worst-case MSE under the Horvitz-Thompson estimator. This leads to an optimization problem over the covariance matrices of the treatment assignment, trading off interference, homophily, and robustness. We frame and solve this problem using two complementary approaches. The first involves formulating a semidefinite program (SDP) and employing Gaussian rounding, in the spirit of the Goemans-Williamson approximation algorithm for MAXCUT. The second is an adaptation of the Gram-Schmidt Walk, a vector-balancing algorithm which has recently received much attention. Finally, we evaluate the performance of our designs through various experiments on simulated network data and a real village network dataset.

Estimating Item Difficulty with Large Language Models as Experts

Diana Kolesnikova, Kirill Fedyanin, Abe D. Hofman +2 more · 2026

Accurate estimates of item difficulty are essential for valid assessment and effective adaptive learning. However, for newly created tasks, response data are typically unavailable. Pretesting and expert judgement can be costly and slow, while machine learning methods often require large labelled training datasets. Recent work suggests that large language models (LLMs) may help. However, there is limited evidence on the elicitation procedures and prompt configurations used to emulate experts for difficulty estimation. This study addresses this gap by evaluating three off-the-shelf LLMs as difficulty raters for newly created items without access to response data. Using an item bank from an online learning system, the study examined 6 domains of primary-school mathematics, with empirical difficulty estimates treated as empirical reference. The study used a full factorial design crossing three factors: judgement format (absolute vs pairwise), decision type (hard decisions vs token-probability-based estimates), and prompting strategy (zero-shot vs few-shot). LLM-derived difficulty estimates were compared with empirical difficulties using Spearman rank correlations. Across domains, LLM-based estimates exhibited moderate to strong positive correlations with empirical item difficulties. For simpler arithmetic tasks, some configurations approached the upper end of the accuracy range reported for human experts in previous research. Pairwise comparison consistently outperformed absolute judgement in the absence of additional refinements. However, when token-level probabilities were incorporated and examples of items with known empirical difficulty were provided, the absolute judgement configuration likewise demonstrated moderate-to-high alignment. The study positions LLMs as a promising tool for initial item calibration and offers insights into effective workflow configuration.

Neyman Jackknife: Design-Based Variance Estimation for Causal Inference under Interference

Bryan Park, Stefan Wager · 2026 · 0 citations

We propose a framework, the Neyman Jackknife, for conservative variance estimation in finite-population causal inference under interference. Our approach provides a general, flexible blueprint that enables conservative variance estimation whenever we are able to recompute our target estimator with some treatment assignments omitted. In classical settings, our approach recovers estimators closely related to the Neyman estimator under SUTVA and the Newey-West HAC variance estimator for time series. Numerical experiments suggest that our general-purpose framework yields variance estimators that can match or even surpass the performance of baselines that were purpose-built for specific applications.

Targeted maximum likelihood estimation of vaccine effectiveness and immune correlates in test-negative design studies with missing data

Leah I. B. Andrews, Lars van der Laan, Peter B. Gilbert · 2026

The test-negative design (TND) is a resource-efficient observational study design that can assess vaccine effectiveness and exposure-proximal immune correlates of disease. The TND enrolls symptomatic individuals seeking diagnostic testing and compares case status by an exposure variable, such as vaccination status or immune marker level, that is measured at testing. While the TND reduces confounding by healthcare-seeking behavior, other sources of confounding may remain. TND studies may also have missing data in the exposure variable due to incomplete records or two-phase sampling designs. We present a targeted maximum likelihood estimation approach involving a semiparametric logistic regression model that targets a causal conditional risk ratio of symptomatic disease in the healthcare-seeking population. Under causal and missing at random assumptions, our method produces an efficient, asymptotically linear estimator that provides flexible, data-driven confounding control and valid causal inference when analyzing TND studies with missing exposure variable data. We evaluate our method's finite sample properties using plasmode simulations of a two-phase TND immune correlates study. We also apply our method to assess COVID-19 vaccine effectiveness and antibody marker correlates of COVID-19 from TND study cohorts derived from the Moderna Coronavirus Efficacy phase 3 trial.

Goal-Oriented Lower-Tail Calibration of Gaussian Processes for Bayesian Optimization

Aurélien Pion, Emmanuel Vazquez · 2026

Bayesian optimization (BO) selects evaluation points for expensive black-box objectives using Gaussian process (GP) predictive distributions. Kernel choice and hyperparameter selection can lead to miscalibrated predictive distributions and an inappropriate exploration-exploitation trade-off. For minimization, sampling criteria such as expected improvement (EI) depend on the predictive distribution below the current best value, so lower-tail miscalibration directly affects the sampling decision. This article studies goal-oriented calibration of GP predictive distributions below a low threshold $t$ in the noiseless setting, for standard GP models with hyperparameters selected by maximum likelihood. A framework for predictive reliability below $t$ is introduced, based on two notions of spatial calibration: occurrence calibration over the design space and thresholded $μ$-calibration on sublevel sets of the form $\{x\in\mathbb{X}, f(x)\le t\}$. Building on this framework, we propose tcGP, a post-hoc method that calibrates GP predictive distributions below~$t$, and we show that the resulting EI-based global optimization algorithm remains dense in the design space. Experiments on standard benchmarks show improved lower-tail calibration and BO performance relative to standard GP models and globally calibrated GP models.

CausalGuard: Conformal Inference under Graph Uncertainty

Vikash Singh, Weicong Chen, Debargha Ganguly +12 more · 2026

Estimating treatment effects from observational data requires choosing an adjustment set, but valid adjustment depends on an unknown causal graph. Graph misspecification can cause under-coverage, while graph-agnostic conformal wrappers may regain nominal coverage only through large padding. We introduce CausalGuard, a structure-weighted conformal framework that calibrates after aggregating graph-conditional doubly robust pseudo-outcomes. Candidate DAGs are proposed from an LLM-derived edge prior, pruned by conditional-independence tests, and reweighted by Bayesian Information Criterion. A composite nonconformity score then calibrates the posterior-weighted pseudo-outcome. CausalGuard provides distribution-free finite-sample marginal coverage for this aggregated pseudo-outcome; under causal identification, overlap, conditional-mean nuisance stability, and concentration on target-aligned valid adjustment strategies, its conditional mean converges to the true Conditional Average Treatment Effect. Across five benchmarks, CausalGuard attains mean coverage above the nominal 90% level for the directly evaluable target and reduces width when graph-agnostic conformal baselines require large padding. Stress tests show that CausalGuard suppresses invalid collider adjustment and remains stable under misspecified priors when the retained candidate set is data-supported.

A Scalable Parametric Item Calibration Engine (SPICE) for Explanatory IRT with Sparse Data

Steven W. Nydick, Manqian Liao, J. R. Lockwood · 2026

We describe a Bayesian multidimensional explanatory IRT model, and an associated Markov Chain Monte Carlo (MCMC) estimation procedure and the corresponding development of calibration software, designed for psychometric analyses of large numbers of sparsely-linked persons and items. Such data structures can arise, for example, from adaptive assessments using large banks of automatically generated items with individual test takers receiving a very small proportion of the entire bank. We discuss how our choices for model specification, data structures, and algorithm implementation combine to create a scalable method for explanatory IRT that can support a variety of psychometric operations with sparse data.

Active Context Selection Improves Simple Regret in Contextual Bandits

Mohammad Shahverdikondori, Jalal Etesami, Negar Kiyavash · 2026

We study the contextual multi-armed bandit problem with a finite context space (a.k.a. subpopulations), where the learner recommends a best action for each context and is evaluated by context-weighted simple regret. Our guarantees are worst-case over the reward distributions, while remaining instance-dependent with respect to the context distribution vector $p$. Akin to experimental design problems where the population of interest is fixed but the sampled subpopulation can be controlled, we allow the learner to actively choose which context to sample from. For a known $p$, we characterize tight regret rates: passive sampling where contexts are randomly revealed achieves regret of order $\sqrt{n/T \, \lVert p \rVert_{1/2}}$, whereas active sampling with allocation $q_j \propto p_j^{2/3}$ achieves the tight rate $\sqrt{n/T} \, \lVert p \rVert_{2/3}$. The resulting improvement can be as large as $Θ(k^{1/4})$, where $k$ is the number of contexts. We further extend the analysis to budgeted active sampling, characterize the corresponding tight rate, and identify when a limited active budget suffices to recover the fully active rate. When $p$ is unknown, we propose the Explore-Explore-Then-Commit (EETC) algorithm, which optimally balances estimating the context distribution and the time to switch to active allocation, such that for large horizons, it matches the known-$p$ active rate up to constants. Experiments on synthetic and real-world data support our theoretical findings.

Individualized Causal Effects under Network Interference with Combinatorial Treatments

Yunping Lu, Haoang Chi, Qirui Hu +1 more · 2026

Modern causal decision-making increasingly demands individualized treatment-effect estimation in networks where interventions are high-dimensional, combinatorial vectors. While network interference, effect heterogeneity, and multi-dimensional treatments have been studied separately, their intersection yields an exponentially large intervention space that makes standard identification tools and low-dimensional exposure mappings untenable. We bridge this gap with a unified framework that constructs a \emph{global potential-outcome emulator} for unit-level inference. Our method combines (1) rooted network configurations to leverage local smoothness, (2) doubly robust orthogonalization to mitigate confounding from network position and covariates, and (3) sparse spectral learning to efficiently estimate response surfaces over the $2^p$-dimensional treatment space. We also decompose networked effects into own-treatment, structural, and interaction components, and provide finite-sample error bounds and asymptotic consistency guarantees. Overall, we show that individualized causal inference remains feasible in high-dimensional networked settings without collapsing the intervention space.

On the Impossibility of Specification Testing of Interference Models Based on Exposure Mappings

Chao Gao, Christopher Harshaw, Fredrik Sävje +1 more · 2026

In order to estimate causal effects in a randomized experiment where spillovers are suspected to occur, analysts must posit a model of interference. The most popular class of interference models are those based on exposure mappings. In practice, it is rarely clear which interference model accurately captures the true nature of spillovers in the experiment. In response, researchers have developed specification tests which seek to determine whether a given interference model is correctly specified. In this context, Type I error is the rejection rate when the interference model is actually correct and Type II error is the acceptance rate when the interference model is incorrectly specified. While existing tests have been explicitly constructed to control Type I error, their Type II error remains less well understood. In this paper, we provide a strong impossibility result: any specification test for an exposure mapping model which aims to have power against a larger exposure mapping model has worst-case Type I and Type II errors that sum to one. This means that no specification test can provide uniformly better performance than the naive test which discards all data and rejects the null at random. Our negative result holds for all sample sizes, for uniformly bounded outcomes, and for alternatives which are maximally separated from the null. Informative specification tests must therefore further restrict the alternative model against which they seek to attain power. To this end, we provide a uniformly consistent test for differentiating no-interference from a network-linear-in-means model.

Vibe Econometrics and the Analysis Contract

Lydia Ashton · 2026 · 0 citations

"Vibe coding" and "vibe analytics" have been framed as a democratization of technical capability. This paper argues that AI-assisted methodology more broadly, or what I call "vibe methodology," also democratizes the failure modes specific to each domain. When AI assists with methods whose validity depends on assumptions that cannot be verified from the output alone (a class I call "vibe inference"), the failure surface is structurally different: the output does not reliably signal invalidity, and when it does, recognizing the signal requires the expertise the workflow bypasses. I focus on "vibe econometrics," the subset of AI-assisted causal analysis where identification can be named faster than it can be audited. The claim of this paper is not that AI invents inferential failures that did not previously exist, but that it changes their incidence, observability, and persuasive force enough to create a practically distinct governance problem. This results in three failure modes: method-data mismatch, where AI bypasses expertise at execution; confidence laundering, where AI amplifies the credibility of formatted output; and invisible forking, which spans both. What is new is not the failure modes but AI's industrialization of their packaging. The barrier between naming a method and executing it has collapsed, and weak foundations, dressed as rigorous analysis, now reach audiences at a scale, speed, and polish that previously required expertise. I propose the Analysis Contract, a pre-commitment framework that adapts the logic of pre-analysis plans and the Causal Roadmap to the AI-assisted setting. The contract imposes three conditions before a causal claim is made: a method-data contract, a data audit, and a pre-commitment statement defining what would count as a disconfirming result. The framework generalizes across domains of vibe inference through domain-specific instantiation.

Selecting Informative Conformal Prediction Sets with an Optimized FCR-Controlled Approach

Israela Solomon, Etienne Roquain, Saharon Rosset +1 more · 2026

Conformal methods provide prediction sets for outcomes with confidence guarantees. We study their use in a selective inference setting, where inference is performed only when the prediction set is informative. The analyst may consider as informative, for example, cases with prediction sets that are sufficiently small, exclude null values, or satisfy other appropriate monotone constraints. Because inference is typically restricted to informative cases in practical applications, accounting for the resulting selection bias is crucial to maintaining false coverage rate (FCR) control. A general framework for constructing such informative conformal prediction sets while controlling the FCR on the selected sample was suggested in Gazin et al. (2025). In this work we focus on oracle-guided procedures. We derive the optimal decision policy under a suitable power objective in the oracle setting where the probability of belonging to each prediction set can be computed. In practice, of course, only estimated probabilities are available. We therefore introduce a calibration procedure that adjusts the oracle policy to maintain finite sample FCR control. We show that this approach can achieve substantially higher power than available alternatives. We demonstrate the effectiveness of our new methods for classification outcomes on both real and simulated data.

Controlling False Discovery in Arbitrarily Structured Hypothesis Spaces via Reproducing Kernels

Binyamin Perets, Shie Mannor · 2026

Large-scale hypothesis testing is central to modern science, where controlling the False Discovery Rate (FDR) has become the standard approach to managing false positives across many simultaneous tests. Hypotheses rarely exist in isolation; they often exhibit structure through proximity, connectivity, or hierarchy. This structure represents both a challenge and an opportunity: while classical methods treat these dependencies as obstacles requiring conservative correction, leveraging them can substantially increase discovery power. Here, we reframe structured FDR control as a regularized learning problem. By optimizing within a suitable Reproducing Kernel Hilbert Space (RKHS), we introduce a framework that unifies continuous domains, graphs, and hierarchies under a single algorithm through kernel choice alone. This formulation enables smooth solutions in place of the piecewise-constant fits of prior methods, principled likelihood-based hyperparameter selection rather than heuristic tuning, and inference at unobserved locations which in turn supports sample-efficient experimental design. Building on this estimator, we provide two decision rules which we prove to control the FDR. We validate our method on two sources: spatial locations derived from high-dimensional real-world datasets, and a differential gene expression task utilizing protein-protein interaction graphs.

Nonparametric Bayesian Policy Learning

Haonan Ye · 2026 · 41 citations

I propose Nonparametric Bayesian Policy Learning (NBPL) as a framework for uncertainty-aware treatment choice. I consider a decision-maker (DM) seeking to select an expected welfare-maximizing treatment rule using observable characteristics. A key observation is that, for a given welfare criterion and policy class, uncertainty about welfare-relevant objects is entirely induced by uncertainty about a reduced-form distribution. I assume the DM places a nonparametric Dirichlet process prior on this reduced-form parameter and uses the resulting posterior to conduct inference on optimal treatment assignments, optimal welfare, and comparisons across policy classes. The NBPL framework is flexible, and its implementation via the Bayesian bootstrap is highly tractable. I establish two main theoretical properties of NBPL. First, posterior welfare regret under NBPL converges at the minimax-optimal rate. Second, posterior model comparison across policy classes is pointwise consistent. I illustrate NBPL in two empirical applications: the bednet subsidy experiment of Bhattacharya and Dupas (2012) and the JTPA experiment studied by Kitagawa and Tetenov (2018).

Linear models for causal inference under network interference

Eric Tong, Salvador V. Balkus · 2026 · 0 citations

In causal inference, interference occurs when the treatment of one unit may affect the outcomes of other units. The goal of this work is to serve as a guide to the use of linear outcome modeling for estimating causal effects in settings where interference may pose a challenge to identification and estimation, such as spatial and network data. We demonstrate that, under a linear model, causal effects of binary and continuous treatments can be identified in terms of regression coefficients under totally and partially known interference structures. Our work constructs unbiased and consistent point and variance estimators for these effects under one or more possible fixed or random interference networks. A chief advantage is that this approach can be implemented using standard linear regression software, and is easily augmented with random effects and heteroscedastic or autocorrelation consistent standard errors. Numerical experiments and an example data analysis demonstrate the efficacy of this approach in eliminating interference bias.

Chained Markov melding using divide and conquer sequential Monte Carlo

Yixuan Liu, Robert J. B. Goudie · 2026

Specifying a full Bayesian model that integrates multiple data sources can be challenging. One natural approach is to specify each individual model separately and join them afterwards. This is the approach adopted in Markov melding. However, when adjacent submodels share common quantities, as in chained Markov melding, posterior inference can be challenging for existing MCMC-based approaches. In this paper, we propose a new multi-stage sampler for chained Markov models involving an arbitrary number of submodels. The proposed sampler adopts a divide-and-conquer sequential Monte Carlo approach for the tree-structured model that fits naturally with the structure of chained Markov melding. The resulting multi-stage sampler provides a flexible alternative for sampling from complex joint models, as its separate sampling scheme for different submodels avoids the need for directly sampling from the full model. We demonstrate applications of the sampler through two examples. The first is a toy example involving 11 submodels of various types. The second example considers an ecologically integrated population model that combines multiple datasets to estimate immigration and reproduction rates.

Distribution-free root cause analysis

Rohan Hore, Aaditya Ramdas · 2026

We study distribution-free root cause analysis in multi-stream data, where an evolving underlying system is observed through multiple data streams that may each undergo distributional changes at unknown timepoints. In such settings, the stream exhibiting the earliest change provides a natural starting point for investigating the underlying cause, which we refer to as the root-cause index. Leveraging conformal $p$-values, we propose a novel framework, Conformal Root Cause Analysis (CROC), which constructs finite-sample valid confidence sets for the root-cause index under minimal assumptions: the data streams are independent, and within each stream the pre- and post-change observations are sampled exchangeably from arbitrary and unknown distributions. We further establish a universality property, showing that any distribution-free method for root cause localization can be represented within the CROC framework. In addition, under mild regularity conditions and principled score design, our method yields asymptotically sharp confidence sets that efficiently isolate the root cause. We further extend CROC to efficiently handle cross-stream dependence when present. Extensive simulations demonstrate accurate localization of the root stream, supporting our theoretical guarantees.

Everywhere Valid Bounds on False Discovery Proportions in Conformal Inference

Ziang Song, Ying Jin, Emmanuel J. Candès · 2026

Modern applications of conformal inference to multiple testing problems, such as outlier detection and candidate selection, often involve selecting test samples whose conformal p-values fall below a threshold. The quality of such methods is often measured by the false discovery proportion (FDP), defined as the fraction of incorrect selections. Existing approaches typically control the expected value of the FDP, using methods such as the Benjamini-Hochberg procedure. This approach fails to provide high-probability bounds on the realized false discovery proportion and invalidates statistical guarantees if the rejection threshold is selected after inspecting the data. This paper establishes finite-sample, distribution-free upper bounds on the FDP that hold simultaneously over all possible rejection thresholds, enabling arbitrary post hoc selection of the threshold. Simultaneous validity is achieved by constructing a high-probability envelope for the empirical distribution function of null conformal p-values by sampling from their joint distribution. Furthermore, our framework allows practitioners to modulate the envelope's shape, thereby producing tight bounds in rejection regions of primary interest. We use this flexible approach to derive simultaneous FDP upper bounds for both outlier detection and conformal selection. We demonstrate through synthetic and real-data experiments that the resulting bounds are both valid and substantially less conservative than those derived from existing approaches.

Fair Policy Learning under Bipartite Network Interference: Learning Fair and Cost-Effective Environmental Policies

Raphael C. Kim, Rachel C. Nethery, Kevin L. Chen +1 more · 2026

Numerous studies have shown the harmful effects of airborne pollutants on human health. Vulnerable groups and communities often bear a disproportionately larger health burden due to exposure to airborne pollutants. Thus, there is a need to design policies that effectively reduce the public health burdens while ensuring cost-effective policy interventions. Designing policies that optimally benefit the population while ensuring equity between groups under cost constraints is a challenging statistical and causal inference problem. In the context of environmental policy this is further complicated by the fact that interventions target emission sources but health impacts occur in potentially distant communities due to atmospheric pollutant transport -- a setting known as bipartite network interference (BNI). To address these issues, we propose a fair policy learning approach under BNI. Our approach allows to learn cost-effective policies under fairness constraints even accounting for complex BNI data structures. We derive asymptotic properties and demonstrate finite sample performance via Monte Carlo simulations. Finally, we apply the proposed method to a real-world dataset linking power plant scrubber installations to Medicare health records for more than 2 million individuals in the U.S. Our method determine fair scrubber allocations to reduce mortality under fairness and cost constraints.

Nonparametric Empirical Bayes Confidence Intervals

Zhen Xie · 2026

Empirical Bayes methods can improve inference on unobservable individual effects by borrowing strength across units. This paper proposes nonparametric empirical Bayes confidence intervals (NP-EBCIs) for unobservable individual effects in a normal means model. The oracle intervals are constructed from posterior quantiles under a point-identified, fully nonparametric prior; feasible intervals replace these quantiles with nonparametric estimates. The NP-EBCIs are asymptotically exact in the sense that both their conditional and marginal coverage probabilities converge to the nominal level. The flexibility of this nonparametric construction has an unavoidable statistical cost. We demonstrate that posterior quantiles, unlike posterior means, inherit the severe ill-posedness of nonparametric deconvolution: the minimax optimal estimation rate is logarithmic. This logarithmic rate is minimax optimal for errors in the conditional coverage probability, and the resulting errors in the marginal coverage probability also vanish at the same logarithmic rate. Despite these slow asymptotic rates, simulations show that the NP-EBCIs remain close to nominal coverage when the prior is non-Gaussian, and deliver substantial length reductions relative to intervals that treat each unit in isolation.

Component over Composite: Mitigating Type I Error Inflation when Imputing "Days Alive and at Home"

Mia S. Tackney, Sarah Dawson, Letao Yuan +2 more · 2026

Background: Days Alive and at Home (DAH) over a pre-defined follow-up period is a novel post-intervention composite outcome that combines data from at least three components: (i) initial length of hospital stay, (ii) length of total readmissions or other post-discharge care and (iii) mortality. Missing values bring unique challenges to the analysis of trials with the DAH outcome as the three components may have different rates of missingness caused by distinct missing data mechanisms. Current approaches define DAH as missing if any of the components are missing, and proceed with complete cases or Multiple Imputation (MI) of the composite. Methods: Through a simulation study motivated by the NOTACS trial, we compare several methods of handling missing data, including complete case analysis, MI of the composite, and MI of the components when the primary analysis is a Mann-Whitney-Wilcoxon test. Results: MI on the component level has good properties in terms of type I error control and power. We caution against the use of MI on the composite level with Predictive Mean Matching, which can lead to type I error inflation. Conclusions: Given the complex distributional characteristics of DAH, naive approaches such as defining missingness on the composite level and directly imputing the composite with Predictive Mean Matching, can lead to type I error inflation. Imputing on the component level is recommended, suggested future work included imputation approaches that are compatible with more complex definitions of DAH, as well as recommendations for sensitivity analyses to the Missing at Random assumption.

A Grid-Rate Condition for Valid Uniform Inference

Emmanuel Selorm Tsyawo · 2026

Estimating a continuous functional $F: \X \to \R$ involves specifying $L_n^d$ nodes on $\X \subset \R^d$ for estimation and uniform inference. While asymptotically valid inference requires $L_n$ to increase with $n$, existing fixed-$L$ rules of thumb and heuristic data-driven approaches lack formal justification. This paper shows that, for functions within a Donsker class, the simple grid-growth condition \(L_n=ω(r_n^{1/4})\) is sufficient for valid inference for twice continuously differentiable functions estimable at the \(r_n^{1/2}\) rate. This condition ensures that the approximation error is asymptotically negligible relative to the stochastic variation of the empirical process.

1-truncated C-vine copula mixed models for network meta-analysis of multiple diagnostic tests

Aristidis K. Nikoloulopoulos · 2026

As meta-analysis of multiple diagnostic tests impacts clinical decision making and patient health, there is growing interest in statistical models that synthesize evidence from studies comparing multiple diagnostic tests. To compare the accuracy of multiple diagnostic tests in a single study, three designs are commonly used: (i) the multiple test comparison design; (ii) the randomized design, and (iii) the non-comparative design. Generalized linear mixed models (GLMMs) are currently the recommended approach for jointly meta-analyzing data from all three designs, enabling simultaneous inference. In this context, 1-truncated C-vine copula mixed models are proposed as a flexible and powerful alternative. These models generalize the GLMM framework by allowing for arbitrary univariate distributions of the random effects and capturing tail dependencies and asymmetries. We demonstrate the utility of our methods with an extensive simulation study and by insightfully re-analysing a case study on the network meta-analysis of diagnostic tests for deep vein thrombosis. Findings indicate that 1-truncated C-vine copula mixed models can offer improvements over GLMMs, supporting their adoption for network meta-analysis of multiple diagnostic tests.

CASCADE Conformal Prediction: Uncertainty-Adaptive Prediction Intervals for Two-Stage Clinical Decision Support

Ricardo Diaz-Rincon, Muxuan Liang, Adolfo Ramirez-Zamora +1 more · 2026

Effective medication management in Parkinson's Disease (PD) is challenging due to heterogeneous disease progression, variable patient response, and medication side effects. While AI models can forecast levodopa equivalent daily dose (LEDD) as a measure of medication needs, standard uncertainty quantification often fails to communicate the reliability of these predictions, treating high and low confidence clinical decisions identically. We introduce CASCADE (Calibrated Adaptive Scaling via Conformal And Distributional Estimation), a novel conformal prediction framework that propagates epistemic uncertainty from a screening classifier to adapt downstream predictions. Unlike standard conformal methods that rely on auxiliary residual regression, we leverage epistemic uncertainty from a primary classification task (identifying whether a medication change is needed) to dynamically scale the prediction intervals of a secondary regression task (predicting how much change). By mapping Venn-Abers multi-probabilistic uncertainty directly to non-conformity scores, our framework achieves continuous risk adaptation. We demonstrate that this ``cascade effect'' produces highly efficient intervals for confident patients (38.9% narrower than standard conformal baselines) while automatically expanding intervals to ensure robust coverage for uncertain cases, bridging the gap between discrete clinical decision-making and continuous dose forecasting in PD.

Compensator-Based Inference for Signal Detection Under Unknown Background

Aritra Banerjee, Sara Algeri · 2026

The problem of detecting new signals in the presence of an unknown background is ubiquitous in scientific discoveries and is especially prominent in the physical sciences. Most solutions proposed thus far to address the problem focus on estimating the background distribution and using that estimate to infer the signal. By studying the geometry of the problem, this article demonstrates that estimating the background distribution is somewhat unnecessary for inferring the signal intensity. Instead, it suffices to estimate a single parameter, referred to as the compensator, to account for the incomplete knowledge on the background, substantially simplifying the problem's complexity and enabling proper uncertainty propagation. Such a compensator is shown to govern the conservativeness of the inference, both in the proposed setup and in likelihood-based approaches.

Low-rank Covariate Balancing Estimators under Interference

Souhardya Sengupta, Kosuke Imai, Georgia Papadogeorgou · 2025 · 1 citations

A key methodological challenge in observational studies with interference between units is twofold: (1) each unit's outcome may depend on many others' treatments, and (2) treatment assignments may exhibit complex dependencies across units. We develop a general statistical framework for constructing robust causal effect estimators to address these challenges. We first show that, without restricting the patterns of interference, the standard inverse probability weighting (IPW) estimator is the only uniformly unbiased estimator when the propensity score is known. In contrast, no estimator has such a property if the propensity score is unknown. We then introduce a \emph{low-rank structure} of potential outcomes as a broad class of structural assumptions about interference. This framework encompasses common assumptions such as anonymous, nearest-neighbor, and additive interference, while flexibly allowing for more complex study-specific interference assumptions. Under this low-rank assumption, we show how to construct an unbiased weighting estimator for a large class of causal estimands. The proposed weighting estimator does not require knowledge of true propensity scores and is therefore robust to unknown treatment assignment dependencies that often exist in observational studies. If the true propensity score is known, we can obtain an unbiased estimator that is more efficient than the IPW estimator by leveraging a low-rank structure. We establish the finite sample and asymptotic properties of the proposed weighting estimator, develop a data-driven procedure to select among candidate low-rank structures, and validate our approach through simulation and empirical studies.

Correlated Random Coefficient Distributions in Linear Panel Models

Irene Botosaru, James L. Powell · 2026

We consider a static linear panel model with both correlated and uncorrelated random coefficients, where the former can depend arbitrarily on observable regressors while the latter are independent of them. We provide sufficient conditions for identification of the distributions of the random coefficients without imposing restrictions on the time-series structure of the error terms in short panels. Our framework applies to regular and irregular designs. The distribution of the correlated coefficients follows via a deconvolution argument. In irregular designs, identification relies on a stayer-based argument exploiting near-singular realizations of the regressor matrix. We develop a two-step minimum distance sieve estimator, with tuning parameters selected by cross-validation. In an application to calorie-expenditure elasticities using data from the randomized evaluation of a conditional cash transfer program, we interpret the estimated distributions by program status as distributions of regime-specific structural calorie-expenditure elasticities. The estimated densities themselves reveal substantial heterogeneity in household-specific elasticities, with nontrivial mass concentrated near zero and a non-negligible share of negative realizations. This heterogeneity implies that responses to income or expenditure changes are not uniformly positive and vary widely across households. Taken together, these features support a framework in which households adjust along both quantity and quality margins, rather than conforming to a homogeneous Engel-curve response.

Assessing covariate-adjusted risk differences in small-sample clinical trials

Martin Schnuerch, Alex Ocampo, Klaus Kähler Holst +1 more · 2026

Binary endpoints are common in clinical trials and conditional odds ratios have traditionally been used to assess treatment effects. However, the interpretation of odds ratios is difficult, they are non-collapsible and rely on strong assumptions in order to be a relevant overall summary measure for the trial. As an alternative, risk differences have gained increasing prominence as a more interpretable, clinically meaningful and assumption-lean measure of treatment effects. This shift has also been motivated by new regulatory guidance, which emphasizes the relevance of marginal estimands and encourages covariate adjustment. Yet, covariate-adjusted inference for risk differences, particularly in smaller samples, has methodological subtleties and lacks well-established best practices. We conduct a simulation study comparing methods for estimating and testing risk differences in small-sample ($N \leq 150$) randomized clinical trials with prognostic categorical baseline covariates, focusing on exact unconditional tests, Mantel-Haenszel methods, and $g$-computation (standardization) approaches. We find that several $g$-computation approaches exhibit inflated Type-I error in very small samples when standard Wald-type inference is applied, whereas robust or penalized variants improve error control at the expense of power. Classical methods such as the Mantel-Haenszel and Suissa-Shuster tests remain robust but may forgo efficiency gains from covariate adjustment. Overall, our results indicate that much of the observed Type-I error inflation reflects misalignment between estimand and variance estimation rather than small sample size alone. Based on these results, we provide practical recommendations to guide method selection that align the estimand, variance estimation, and inferential target.

Endogenous Quantile Regression with Measurement Error in Dependent Variable

Xuanjing Su · 2026

This paper studies quantile regression with an endogenous regressor and measurement error in the dependent variable. Standard quantile regression estimators ignoring these two elements can induce substantial bias. We adopt a control-function approach in a triangular system and show that the conditional quantile coefficient functions, together with all other distributional parameters, are nonparametrically identifiable. Building on this constructive identification result, we propose a two-step sieve ML estimator. The first step estimates the control function. The second step performs a sieve likelihood maximization that incorporates the generated control variable through copula weights. When the number of quantile grid knots grows at an appropriate speed, the estimator is consistent and asymptotically normal, permitting inference via bootstrap. Monte Carlo simulations demonstrate that the estimator markedly reduces bias relative to existing methods, confirming its effectiveness in settings with endogeneity and additive measurement error in the outcome.

Nonparametric efficient inference for network quantile causal effects under partial interference

Chao Cheng, Fan Li · 2026

Interference arises when the treatment assigned to one individual affects the outcomes of other individuals. Commonly, individuals are naturally grouped into clusters, and interference occurs only among individuals within the same cluster, a setting referred to as partial interference. We study network causal effects on outcome quantiles in the presence of partial interference. We develop a general nonparametric efficiency theory for estimating these network quantile causal effects, which leads to a nonparametrically efficient estimator. The proposed estimator is consistent and asymptotically normal with parametric convergence rates, while allowing for flexible, data-adaptive estimation of complex nuisance functions. We leverage a three-way cross-fitting procedure that avoids direct estimation of the conditional outcome distribution. Simulations demonstrate adequate finite-sample performance of the proposed estimators, and we apply the methods to a clustered observational study.

Journey to the Centre of Cluster: Harnessing Interior Nodes for A/B Testing under Network Interference

Qianyi Chen, Anpeng Wu, Bo Li +2 more · 2026

A/B testing on platforms often faces challenges from network interference, where a unit's outcome depends not only on its own treatment but also on the treatments of its network neighbors. To address this, cluster-level randomization has become standard, enabling the use of network-aware estimators. These estimators typically trim the data to retain only a subset of informative units, achieving low bias under suitable conditions but often suffering from high variance. In this paper, we first demonstrate that the interior nodes - units whose neighbors all lie within the same cluster - constitute the vast majority of the post-trimming subpopulation. In light of this, we propose directly averaging over the interior nodes to construct the mean-in-interior (MII) estimator, which circumvents the delicate reweighting required by existing network-aware estimators and substantially reduces variance in classical settings. However, we show that interior nodes are often not representative of the full population, particularly in terms of network-dependent covariates, leading to notable bias. We then augment the MII estimator with a counterfactual predictor trained on the entire network, allowing us to adjust for covariate distribution shifts between the interior nodes and full population. By rearranging the expression, we reveal that our augmented MII estimator embodies an analytical form of the point estimator within prediction-powered inference framework. This insight motivates a semi-supervised lens, wherein interior nodes are treated as labeled data subject to selection bias. Extensive and challenging simulation studies demonstrate the outstanding performance of our augmented MII estimator across various settings.

Difference-in-Differences in the Presence of Unknown Interference

Fabrizia Mealli, Javier Viviens · 2025 · 0 citations

The stable unit treatment value (SUTVA) is a crucial assumption in the Difference-in-Differences (DiD) research design. It rules out hidden versions of treatment and any sort of interference and spillover effects across units. Even if this is a strong assumption, it has not received much attention from DiD practitioners and, in many cases, it is not even explicitly stated as an assumption, especially the no-interference assumption. In this technical note, we investigate what the DiD estimand identifies in the presence of unknown interference. We show that the DiD estimand identifies a contrast of causal effects, but it is not informative on any of these causal effects separately, without invoking further assumptions. Then, we explore different sets of assumptions under which the DiD estimand becomes informative about specific causal effects. We illustrate these results by revisiting the seminal paper on minimum wages and employment by Card and Krueger (1994).

Evaluating causal indirect effects when mediators are left-censored by assay limit of quantification

Cong Jiang, Michael D. Hughes, Nima S. Hejazi · 2026

Causal mediation analysis is essential for disentangling the mechanisms by which investigational therapeutic and preventive agents impact clinical outcomes. However, the measurement of biological mediators is often subject to left-censoring by technical measurement limitations, most commonly an assay's limit of quantification. This form of censoring can pose severe challenges for both identification and estimation of causal mediation estimands, particularly when the censoring mechanism is deterministic and the resulting missingness is missing not at random (MNAR) or nonignorable. Motivated by the question of assessing the role of viral RNA in the action mechanism of monoclonal antibody therapies for COVID-19 in the Accelerating COVID-19 Therapeutics and Vaccine (ACTIV)-2 platform trial, we develop a semi-parametric framework for estimation of the natural direct and indirect effects when the mediator of interest is partially subject to this form of left-censoring. Our proposed strategy combines fractional imputation with a semi-parametric EM algorithm to flexibly estimate key components of the factorized data likelihood. Applying the proposed strategy to circumvent the left-censoring, we discuss both traditional plug-in and asymptotically efficient estimators of the direct and indirect effect estimands, introducing a data-adaptive $m$-out-of-$n$ bootstrap for robust inference under the imputation procedure. We demonstrate in numerical experiments that our approach significantly reduces bias and allows for reliable inference. An application to data from the ACTIV-2 platform trial confirms that monoclonal antibody therapies reduce the risk of hospitalization and death due to COVID-19, while suggesting that changes in viral RNA mediate only a modest proportion of the overall treatment effect.

How does limma-trend work? An empirical partially Bayes perspective

Sagnik Nandy, Wanyi Ling, Nikolaos Ignatiadis · 2026

In high-throughput biology, it is common to fit thousands of linear regressions -- one per gene, protein, or other unit -- with very few samples per unit. Limma-trend, one of the most widely used methods in this setting, improves power by shrinking variance estimates parametrically toward a fitted curve (the trend) relating variance to a unit-level summary (e.g., average intensity, peptide count), before computing p-values and applying the Benjamini-Hochberg procedure to control the false discovery rate (FDR). We study limma-trend through the lens of empirical partially Bayes inference, a paradigm in which a prior is posited and estimated for the nuisance parameters while parameters of interest remain fixed. From this perspective, limma-trend computes approximate partially Bayes p-values that condition on the residual sample variance and the unit-level summary. The same framework explains why MAnorm2, a popular variant for ChIP-seq, can sometimes fail to control FDR. We then derive a nonparametric generalization of limma-trend that estimates the residual variance prior using nonparametric maximum likelihood. Under dense signals, this procedure asymptotically controls the FDR -- even when the trend is misspecified or inconsistently estimated. To allow the full shape of the conditional variance distribution to depend on the unit-level summary, we develop a second procedure that learns it directly.

Clustering Craters on the Moon with Dysfunctional Families

Nathan Weed, Emily Castleton, Dave Osthus +2 more · 2026

Summaries of craters on terrestrial bodies, such as the number and size distribution, are essential for understanding the history of the Solar System. Identifying craters, however, has not been automated and thus relies on expert crater-counters marking static images. Robbins et al. (2014) (hereafter R14) showed that, contrary to previously held assumptions, there exists large variability across expert crater-counters' identified crater lists. How best to combine identified crater lists across multiple experts for the purposes of learning about the Solar System is an open and consequential question. R14 combined identified crater lists via clustering through a modification of the popular DBSCAN clustering method. Their approach did not, however, make use of all the constraining information available nor did it provide an estimate of clustering uncertainty. To address the shortcomings of the DBSCAN method, we present a novel clustering approach that can combine multiple lists of identified objects of interest from the same image. The key innovation is incorporating a dysfunctional family constraint into the Bayesian nonparametric clustering approach, the Chinese restaurant process (CRP), which naturally takes into account information about the crater identifier. The dysfunctional family Chinese restaurant process (DFCRP) provides an estimate of clustering uncertainty. In this work, we provide guidance on hyperparameter specification, present a Gibbs sampler, and perform a simulation study to compare the performance of the DFCRP to the CRP. Finally, we apply the DFCRP to the crater identification problem of R14, comparing results, and also demonstrate the types of analyses that can be performed with posterior draws of cluster assignments.

Optimal Sampling for Kernel Quadrature on Unbounded Domains

Edoardo Bandoni, Christian Robert, Julien Stoehr · 2026

Kernel quadrature is widely used to approximate integrals of smooth functions, with worst-case error typically decaying at the minimax rate $n^{-α/d}$ for smoothness $α$ in dimension $d$. Existing rate-optimal methods often depend on deterministic point sets tailored to a specific kernel, making them sensitive to misspecification and less robust in practice. In this work, we study randomized quadrature methods with a focus on robustness rather than kernel-specific optimality. We construct an explicit, $n$-dependent sampling distribution that achieves minimax rates for worst-case error over smoothness classes without requiring knowledge of the kernel. This kernel-agnostic design improves robustness while retaining optimal rates. Our analysis includes unbounded sampling measures such as Gaussian and Student-$t$ distributions, extending beyond compact domains. The results provide both theoretical guarantees and a practical recipe for robust, rate-optimal randomized quadrature.

Evaluation of the number of clusters in a data set using $p$-values from Multiple Tests of Hypotheses

Soumita Modak · 2026 · 4 citations

This paper proposes a novel, nonparametric, interpoint distance-based measure to investigate whether there exist any groups in a set of given data, and if so then, how many groups are prevailing in total. It is a cluster accuracy index useful for arbitrary-dimensional data set, in association with any clustering algorithm having the number of groups specified as a priori. We perform univariate, nonparametric, multiple statistical tests of hypotheses, where as many dependent tests as the sample size are carried out using the interpoint distances. They possess $p$-values to be combined to reach a decision, which is taken in a step-wise process for a possible number of clusters. It reduces the unnecessary computations compared with the other accuracy measures from the literature. Data study establishes the proposed index's efficiency and superiority.

Two-stage Ensemble Clustering of Functional Data Using Random Projections

Sourav Chakrabarty, Anirvan Chakraborty, Shyamal K. De · 2026

We propose a computationally simple framework for clustering functional data based on Gaussian-process-generated random projections. In this approach, each curve is first projected onto a large collection of independent Gaussian process realizations. The resulting high-dimensional representations are clustered using the Mean Absolute Difference of Distances (MADD), a dissimilarity measure well suited for high-dimensional settings. A population-level analysis of this dissimilarity provides insight into how random projections help capture distributional differences between functional populations. We introduce a second stage of clustering to additionally leverage on data-driven projection directions. Thus, in Stage I, an initial clustering is obtained using a set of prespecified projection families. In Stage II, this partition is refined by constructing Gaussian random projections based on an estimated covariance operator that uses the first stage of cluster labels. Finally, a normalized cost function is used to select the optimal clustering among candidate solutions. The proposed clustering algorithm is broadly applicable to diverse functional data regimes including irregular and partially observed data. Through extensive simulations and real-data applications, we show that the proposed method achieves a high degree of accuracy and outperforms many of the state-of-the-art methods across a wide range of functional data settings.

Application of Propensity Score Models and Causal Estimators in Observational Studies under Model Misspecification

Apu Chandra Das, Sakib Salam, Md Robiul Islam Talukder +3 more · 2026

Propensity score (PS) methods are widely used in observational studies to reduce confounding and estimate causal treatment effects. However, the validity of PS-based causal estimators depends heavily on correct model specification, and model misspecification may lead to substantial bias and instability. In this study, we systematically evaluate the performance of commonly used causal estimators, including response surface modeling (RSM), inverse probability weighting (IPW), and augmented inverse probability weighting (AIPW), under varying levels of PS and outcome model misspecification. We compare classical logistic regression with several machine learning approaches for PS estimation, including random forests (RF), support vector machines (SVM), and linear discriminant analysis (LDA). Extensive simulation studies were conducted under multiple scenarios defined by combinations of correctly specified and misspecified PS and outcome models, varying sample sizes, and different covariate correlation structures. Estimator performance was assessed using bias, absolute bias, root mean squared error, empirical standard error, and confidence interval width. Results demonstrate that AIPW consistently provides robust and stable estimates across most scenarios due to its doubly robust property, whereas IPW is highly sensitive to PS misspecification and unstable PS estimates produced by flexible machine learning methods. RSM performs well only when the outcome model is correctly specified. Real-world applications using the ACTG175 clinical trial and the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset further illustrate the practical implications of estimator choice and PS modeling strategy. Overall, our findings highlight the importance of integrating flexible machine learning approaches within doubly robust frameworks to improve causal effect estimation in observational studies.

Robust Simulation Based Inference Through Robust Optimal Transport

Peter Matthew Jacobs, Lekha Patel, Anirban Bhattacharya +1 more · 2026

When a statistical model $\{P_θ : θ\in Θ\}$ lacks analytically tractable likelihoods, parametric statistical inference based on data generated from an unknown underlying distribution $P$ can still be performed as long as simulations from the model are possible. This approach is called Simulation Based Inference (SBI). Statistical models are rarely exactly correct (that is, $P \notin \{P_θ: θ\in Θ\}$), and Robust SBI focuses on inferring a reasonable parameter even under model mis-specification. We focus on the setting where $P$ possesses potentially both geometric and Total Variation type discrepancies from $P_{θ^*}$. For this problem, we use a Kullback-Liebler informed robust Optimal Transport divergence, motivated by Empirical Likelihood considerations. We introduce a stochastic sub-gradient ascent algorithm with a convergence guarantee for estimating the semi-discrete version of this robust Optimal Transport divergence, and design a parallelized SBI algorithm which employs the regular bootstrap on top of minimum semi-discrete robust Optimal Transport for parameter uncertainty quantification. We demonstrate mathematically why the divergence is robust under a joint geometric plus Total Variation type contamination and then illustrate the robustness of inferences on a complex benchmark SBI task.

New Confidence Regions for Linear Regression Parameters with Stationary-Ergodic Dependent Errors

Mous-Abou Hamadou, Martial Longla, Mathias Nthiani Muia +1 more · 2026

We develop joint confidence regions for linear regression coefficients when the regressors and errors are jointly stationary and ergodic with unspecified serial dependence. The method applies random smoothing, using an independent auxiliary sample and shrinking bandwidth, to a vector of regression and second-moment statistics. Under stationarity, ergodicity, and finite second moments, the estimator is asymptotically normal and yields Wald confidence regions and simultaneous confidence intervals without direct long-run variance estimation or a parametric dependence model. For implementation, we introduce a scaled estimator with data-driven bandwidth selection and a mild truncation that improves finite-sample stability. Simulations under ARMA, ARFIMA, copula-based Markov errors, and fractional Gaussian noise, with Gaussian and heavy-tailed margins, show near-nominal coverage and competitive region volumes relative to Newey-West HAC and MAC. A winter Beijing PM2.5 application illustrates the procedure. Keywords: Random smoothing, Joint inference, Confidence regions, Dependent errors, Long memory, Regression inference

Disentangling spatial interference and spatial confounding biases in causal inference

Isqeel Ogunsola, Olatunji Johnson · 2026

Spatial interference and spatial confounding are two major issues inhibiting precise causal estimates when dealing with observational spatial data. Moreover, the definition and interpretation of spatial confounding remain arguable in the literature. In this paper, our goal is to provide clarity in a novel way on misconception and issues around spatial confounding from Directed Acyclic Graph (DAG) perspective and to disentangle both direct, indirect spatial confounding and spatial interference based on bias induced on causal estimates. Also, existing analyses of spatial confounding bias typically rely on Normality assumptions for treatments and confounders, assumptions that are often violated in practice. Relaxing these assumptions, we derive analytical expressions for spatial confounding bias under more general distributional settings using Poisson as example . We showed that the choice of spatial weights, the distribution of the treatment, and the magnitude of interference critically determine the extent of bias due to spatial interference. We further demonstrate that direct and indirect spatial confounding can be disentangled, with both the weight matrix and the nature of exposure playing central roles in determining the magnitude of indirect bias. Theoretical results are supported by simulation studies and an application to real-world spatial data. In future, parametric frameworks for concomitantly adjusting for spatial interference, direct and indirect spatial confounding for both direct and mediated effects estimation will be developed.