series expansion methods

This study addresses the unreliability of conventional asymptotic-based statistical inference in meta-analysis when the number of included studies is small. Focusing on the Gaussian random-effects model, the authors derive—for the first time—a rigorous higher-order Bartlett correction formula tailored to small-sample settings, rectifying an error present in the existing literature. Built upon higher-order asymptotic theory, the proposed correction substantially improves the accuracy of the finite-sample distributional approximation of test statistics. Consequently, it enhances the validity and reliability of hypothesis testing in meta-analytic applications, particularly where sample sizes are limited.

This study addresses the computational complexity associated with calculating quantiles of the inverse normal distribution, Student’s t-distribution, and outlier rejection criteria in hypothesis testing. To overcome the reliance on table lookups or iterative numerical methods, the paper proposes concise and highly accurate analytical approximations formulated as closed-form expressions. These approximations significantly reduce computational overhead while maintaining precision sufficient for practical statistical applications. The resulting method offers substantial gains in computational efficiency, making it particularly well-suited for resource-constrained environments or scenarios requiring rapid statistical inference. By bridging theoretical rigor with practical utility, the approach delivers both methodological insight and real-world applicability.

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.

Diffusion models are commonly assumed to rely solely on Gaussian priors, raising fundamental questions about whether higher-order cumulants (i.e., n-point correlation functions) beyond Gaussian statistics are erased during the forward process and whether they can be recovered in the reverse process. Method: The authors combine cumulant generating functional analysis, score estimation theory, analytic derivation, and exactly solvable toy models, complemented by numerical validation in scalar lattice field theory. Contribution/Results: For variance-increasing forward processes, the authors provide the first rigorous proof that all higher-order cumulants are conserved; their information is fully encoded in the score function and precisely reconstructed by the reverse sampling process. Crucially, even when the forward process endpoint is approximately Gaussian, higher-order cumulants remain non-vanishing and are accurately recovered—challenging the conventional view that diffusion models fundamentally depend only on Gaussian assumptions. This establishes a theoretical foundation for modeling non-Gaussian structure in diffusion-based generative modeling.

Gaussian processes (GPs) face prohibitive computational complexity and poor scalability when modeling large-scale spatial data. To address this, we propose the Full-Scale Approximation (FSA) framework, which unifies global and local covariance structures within a single coherent model. We introduce a novel adaptive preconditioner that substantially accelerates convergence and enhances robustness of the conjugate gradient method for solving linear systems arising from GP inference. Additionally, we design an efficient prediction variance estimator based on stochastic trace approximation, circumventing costly matrix inversions. Empirical evaluations on both synthetic and real-world datasets demonstrate that FSA achieves accuracy comparable to exact inference via Cholesky decomposition while reducing computation time by orders of magnitude. The core implementation is provided as an open-source C++ library, with Python and R interfaces for broad accessibility.

This work addresses the lack of a unified and efficient inference framework for models where exact likelihoods are intractable. It proposes the first general modeling and inference framework based on saddlepoint approximation, which preserves access to the moment-generating function through high-level operations to automatically construct the cumulant-generating function, its saddlepoint, and associated gradients. By integrating automatic differentiation, the framework optimizes the saddlepoint likelihood efficiently. It supports flexible modeling with complex distributional compositions and introduces diagnostic metrics that assess approximation error without requiring the true likelihood. Empirical results demonstrate that the method enables accurate parameter estimation, standard error computation, and error diagnosis with high computational efficiency and remarkable flexibility across multiple case studies.

This work addresses the absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.

This work proposes a novel class of Lancaster copulas by integrating, for the first time, the orthogonal expansion of continuous Lancaster probabilities with copula theory. By constructing their infinite series representation and associated density expressions, and by systematically analyzing truncation effects, the authors achieve efficient approximations of both the target copula and its density. Theoretical analysis and numerical experiments demonstrate that high-accuracy approximation can be attained using only low-order truncations, offering both rigorous theoretical guarantees and computational tractability. This approach provides a powerful new tool for modeling complex dependence structures in multivariate data.

This work addresses the fragmentation in existing Gaussian process–based approaches for approximating solutions to differential equations by proposing a unified Bayesian probabilistic framework. The framework embeds differential equation constraints into the likelihood function through derivative matching, thereby enabling simultaneous estimation of unknown parameters and quantification of uncertainty in the solution. It systematically integrates several established Gaussian process methods for the first time, elucidating their underlying connections. The generality and efficacy of the proposed approach are demonstrated across multiple benchmark problems, establishing a coherent foundation for future theoretical advancements and practical applications in this domain.

This work addresses the challenge of excessive variance in computing N-point correlation functions in lattice quantum chromodynamics by introducing, for the first time, a method that combines normalizing flows with the generating functional of quantum field theory. By modeling derivatives of the source operator within this framework, the authors systematically construct low-variance estimators for correlation functions. The approach is universally applicable to arbitrary bosonic operators and asymptotically approaches the theoretically optimal noise-free estimator. Numerical experiments on glueball correlators and Wilson loops demonstrate variance reductions of up to three orders of magnitude, substantially enhancing both computational efficiency and precision.