asymptotic inference

This paper addresses the challenge of constructing confidence sets (CSs) for semiparametric models that simultaneously achieve finite-sample reliability and asymptotic efficiency. We introduce the novel concept of “non-asymptotically valid and asymptotically exact” (NAVAE) CSs and establish sufficient conditions for their existence. Under mild moment conditions—such as bounded kurtosis or weak exogeneity—we construct closed-form NAVAE confidence intervals for linear combinations of expectations and regression coefficients. These intervals moderately widen classical central-limit-theorem-based intervals to ensure non-asymptotic coverage validity, while embedding a uniform asymptotic exactness framework to robustly accommodate heteroskedasticity and weakly exogenous covariates. Simulation studies demonstrate accurate finite-sample coverage and optimal asymptotic convergence rates, substantially outperforming conventional asymptotic methods. Furthermore, we characterize the theoretical limits of the approach under highly skewed distributions, including the Bernoulli case.

Classical algorithms for strongly convex stochastic optimization achieve fast convergence (O(1/√n)) but suffer from asymptotically non-negligible bias, violating the conditions required for a valid central limit theorem (CLT) and thus impeding asymptotically efficient statistical inference. Method: We propose the first dual-objective algorithm that simultaneously guarantees fast convergence and a provable CLT. Our approach integrates stochastic approximation, asymptotic statistical inference, and adaptive experimental design into a unified framework that ensures asymptotic normality of the estimator. Contribution/Results: We establish theoretical guarantees that the algorithm retains the O(1/√n) convergence rate while satisfying the CLT. Numerical experiments demonstrate substantial improvements over existing methods in estimation accuracy, confidence interval coverage, and identification of optimal treatment parameters. The method provides a new paradigm for continuous, parameterized A/B testing in online platforms—balancing optimization efficiency with statistical reliability.

This paper identifies the systematic failure of classical statistical methods—including mean-based inference, principal component analysis (PCA), and asymptotic normality assumptions—under heavy-tailed distributions, particularly in medium-sample-size (medium-*n*) real-world settings where they are routinely misapplied. Methodologically, it challenges the uncritical adoption of Gaussian and stable-distribution assumptions and introduces the “Median Law” theoretical framework, which formalizes fundamental limitations under heavy tails: unreliable sample means, distorted empirical distributions, and degenerate principal components. The approach integrates extreme value theory, generalized stable distribution modeling, robust parametric estimation, and pre-asymptotic analysis. Empirical validation draws on counterexamples from finance, economics, and psychology, supplemented by cross-disciplinary case studies. The core contribution is a foundational rethinking of uncertainty quantification and causal inference: it demonstrates that many canonical “cognitive biases” are, in fact, rational inferences under heavy-tailed probability structures—thereby advocating a paradigm shift in statistical practice from idealized asymptotics to empirically grounded probabilistic modeling.

This paper addresses uncertainty quantification for risk-minimizing estimators in machine learning, overcoming limitations of classical approaches that rely on restrictive distributional assumptions and asymptotic theory. We propose the first general-purpose, finite-sample, distribution-free, and frequentist-valid inference framework applicable to *any* risk minimizer. Our method is grounded in the generalized likelihood ratio test, integrated with empirical process analysis and data-driven tuning, and inherently supports anytime-valid inference. Theoretically, it guarantees exact coverage of confidence sets for *all* finite sample sizes—without asymptotic approximations. Empirically, it consistently outperforms classical asymptotic methods across diverse tasks, demonstrating both high accuracy and strong robustness. This work establishes a new paradigm for model-agnostic statistical inference.

In statistical inference, confidence sets—especially under complex models or small sample sizes—often fail to achieve nominal coverage levels, particularly in likelihood-free inference (LFI) settings. To address this, we propose TRUST and TRUST++, two distribution-free, simulation-based calibration methods that adapt conformal prediction principles to confidence set construction with redundant parameters, thereby establishing the first distribution-agnostic calibration framework for statistical inference. Our methods guarantee finite-sample local coverage and asymptotic conditional coverage, while enabling self-assessment of simulation cost. Theoretically, we prove their robustness against model misspecification and simulation imperfection. Empirically, TRUST and TRUST++ significantly improve coverage accuracy across both tractable and intractable likelihood models, consistently outperforming existing approaches—especially in small-sample regimes.

This study addresses the limitations of conventional inference methods rooted in sampling variability when sample sizes approach the population size. By constructing finite populations with known parameters and leveraging CPU/GPU-accelerated repeated sampling experiments, the authors examine the evolution of the randomization distribution of the sample mean across varying sampling fractions. Integrating finite population theory with numerical precision analysis, they demonstrate that in high-coverage scenarios, estimation error predominantly stems from computational precision and architectural constraints rather than sampling randomness. The findings reveal that sampling variability becomes negligible well before exhaustive enumeration is reached, thereby challenging a foundational assumption of classical inferential statistics and offering a basis for rethinking statistical paradigms in the context of large-scale, near-complete data.

This study addresses the quantification of sources of predictive uncertainty and their contributions to prediction interval width. Building upon the law of total variance, the work proposes several conservative decompositions of posterior predictive variance, systematically characterizing the components of uncertainty and their interdependencies through conditional expectation and conditional variance terms. Experimental evaluations across multiple canonical models demonstrate that the proposed approach effectively identifies the dominant sources of uncertainty and reveals coherent patterns of co-variation among decomposition terms. These insights offer a novel perspective for model assessment and refinement, enhancing interpretability and guiding targeted improvements in predictive reliability.

This study addresses the failure of standard influence function–based inference in finite samples under “near-boundary” settings of semiparametric models, where second-order remainder terms non-negligibly contribute to sampling variance. The authors propose a finite-sample variance decomposition framework that separates influence function variance from remainder-induced variance and establish necessary and sufficient conditions for the consistency of sandwich variance estimators. Building on this framework, they develop two robust variance estimators—the leave-one-unit-out jackknife and a paired-cluster bootstrap—and derive an analytical expression for the interaction between remainder terms and within-cluster correlation in clustered data. Their approach enables valid confidence interval construction near the boundary: the jackknife Wald interval is numerically equivalent to a bias-corrected sandwich estimator and accurately captures the mechanism by which clustering amplifies variance estimation bias.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This study addresses the estimation of parameters of the form θ₀ = E[F_Y⁻¹∘F_Z(X)] in the “changes-in-changes” model, for which existing methods lack theoretical guarantees when variables are unbounded. The authors construct a plug-in estimator based on empirical quantiles and establish its √n-consistency and asymptotic normality under assumptions weaker than those in the current literature. They further propose a novel consistent estimator for the asymptotic variance. The theoretical analysis leverages empirical process theory and plug-in methods for quantile functions. Monte Carlo simulations demonstrate that the proposed variance estimator substantially outperforms existing alternatives, leading to markedly improved inference accuracy.