edgeworth expansions

This paper addresses exact Bayesian hypothesis testing under models with nuisance parameters, aiming to improve the accuracy of posterior shape characterization. Methodologically, it introduces the first unified framework combining third-order asymptotic expansion and skewness correction for univariate posteriors, yielding highly accurate third-order approximations and skew-normal calibrations. It further extends Bayesian discrepancy measures to the multivariate setting by constructing geometrically coherent and statistically robust credible regions via optimal transport theory. Theoretically, it establishes formal connections between these Bayesian regions and frequentist inference—particularly matching priors. Computationally efficient—incurred cost is only marginally higher than first-order Laplace approximations—the proposed approach significantly enhances both testing accuracy and robustness. Empirical evaluations demonstrate consistent superiority over existing asymptotic methods.

This study addresses accuracy and consistency issues in the numerical construction of the Karhunen–Loève expansion (KLE) arising from discretization, quadrature rules, and finite sample sizes. It establishes an algebraic equivalence between the spectral decomposition of the Fredholm integral equation and the singular value decomposition (SVD) of a weighted sample covariance matrix, thereby unifying model-driven and data-driven KLE frameworks. The work innovatively constructs the covariance function on a non-simply-connected three-dimensional toroidal domain using the shortest interior path distance, and implements the approach numerically with unstructured meshes and Gaussian quadrature. Experiments demonstrate that, in a one-dimensional benchmark problem, SVD-based eigenvalue estimates and empirical KL coefficients converge to the theoretical 𝒩(0,1) distribution. In two-dimensional irregular and three-dimensional toroidal domains, the study systematically quantifies the combined influence of discretization strategy, quadrature accuracy, and sample size on KLE reconstruction error.

Generalized posterior credible sets typically lack asymptotic frequentist coverage guarantees. This work proposes calibrating their coverage by estimating coverage probabilities via the bootstrap and selecting a scalar learning rate to achieve the desired nominal level. Leveraging Edgeworth expansions, we show that coverage error arises from two sources: corrections to the sampling distribution of the estimator and posterior-induced adjustments to the center, boundary, and shape of the credible set. We further establish that a scalar learning rate can yield globally valid calibration only when the posterior and sampling covariance matrices are proportional. Under fixed-dimensional asymptotics, regularity conditions, and local identifiability, we prove—using Edgeworth expansions, classical asymptotic theory, and stochastic approximation—that the solution to the bootstrap coverage equation is consistent, revealing that the procedure essentially implements a scale correction tailored to a specific confidence level.

Bayesian posteriors are often skewed, whereas mainstream deterministic approximations—such as Laplace’s method and variational Bayes—rely on symmetric densities (e.g., Gaussians), leading to systematic bias and reduced accuracy. Method: We propose a generic, optimization-free skewness-aware perturbation framework that can be seamlessly integrated with any off-the-shelf symmetric approximation. Our approach constructs analytical perturbations based on skew-symmetric density families, unifying asymptotic expansion and variational analysis. Contribution/Results: We theoretically establish finite-sample accuracy improvement and prove that the asymptotic convergence rate is accelerated by at least a factor of √n. The method is model-agnostic and compatible with diverse symmetric approximation paradigms. Numerical experiments demonstrate substantial gains over standard Gaussian approximations—particularly in moderate-to-small sample regimes and under strong posterior skewness—empirically validating the predicted convergence acceleration and robustness.

Sampling from high-dimensional target distributions with superlinearly growing potentials—such as nonconvex functions that become convex at infinity—remains challenging. Method: We propose two accelerated higher-order Langevin algorithms, aHOLA and aHOLLA, which integrate higher-order discretizations of Langevin dynamics with momentum-based acceleration. Our theoretical analysis leverages local Hölder continuity, convexity at infinity, and dissipativity conditions. Contribution/Results: We establish the first non-asymptotic convergence bound in Wasserstein-1/2 distance for nonconvex settings, achieving a Wasserstein-1 convergence rate of order $1 + q/2$, where $q$ is the Hölder exponent—surpassing existing rates and attaining the best-known rate for nonconvex sampling. Numerical experiments across diverse nonconvex distributions validate the algorithms’ accelerated convergence, stability, and practical efficiency.

This study addresses the challenge of accurately estimating tail dependence and extreme quantiles of Conditional Higher-order Moment (CoHM) risk measures in financial or insurance settings characterized by weak contagion effects, where existing methods fall short. By leveraging the Farlie–Gumbel–Morgenstern (FGM) dependence structure and integrating extreme value theory with second-order regular variation theory, this work establishes the first second-order asymptotic expansion for CoHM, overcoming the precision limitations inherent in first-order asymptotic approaches. The proposed method substantially enhances approximation accuracy at extreme confidence levels, as demonstrated through numerical simulations showing markedly reduced errors at high quantiles. Empirical analysis further confirms its superior performance when applied to real-world insurance claims data.

This work proposes a “cheap studentized bootstrap” that achieves the same high-order coverage accuracy as the conventional studentized bootstrap while drastically reducing computational cost. Traditional studentized bootstrap methods require extensive resampling or analytical standard error calculations, rendering them computationally expensive. The key innovation lies in formally establishing, for the first time, the connection between studentized statistics and the t-distribution, revealing that the degrees of freedom in the t-distribution reflect the amount of resampling computation rather than the original sample size. Building on Edgeworth and Cornish–Fisher expansions together with limiting t-distribution theory, the authors construct a high-order accuracy framework that maintains rigorous theoretical guarantees with only a minimal number of Monte Carlo resamples.

This work addresses the limitations of existing conformal prediction methods, which typically guarantee only marginal coverage and struggle to ensure conditional coverage for heterogeneous test points or subpopulations, while lacking a unified theoretical framework to analyze their asymptotic validity, compare approaches, or extend them to structured data. The paper proposes the first unified theoretical framework tailored for conditional coverage, deriving non-asymptotic bounds on conditional miscoverage via pointwise and Lₚ paths. It systematically characterizes the sources of error underlying asymptotic conditional validity and provides a coherent interpretation of existing methods. Built upon a weighted symmetry formulation, the framework facilitates conditional coverage–oriented model selection, localization under covariate shift, and natural extensions to structured data. Numerical experiments corroborate the theoretical findings, establishing a comparable, extensible, and practically informative paradigm for conditional coverage.

This study addresses the lack of effective second-order self-normalized inference methods for the $k$-th largest coordinate of sums of high-dimensional independent random vectors, a challenge arising because classical extreme value theory does not readily extend to general order statistics. The authors innovatively reframe the problem as one of estimating rare orthant probabilities and, by integrating factorial moments with a weighted inclusion–exclusion principle, extend existing high-dimensional second-order Gaussian and bootstrap approximation theories—previously limited to maxima—to the $k$-th order statistic. They propose a third-moment-matching wild bootstrap and a pre-pivoted double wild bootstrap, which, under moment conditions, variance assumptions, and weak dependence, achieve a second-order Edgeworth expansion with coverage error of order $n^{-1}$, substantially improving inferential accuracy.

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.