order-statistics analysis

This study addresses the challenge of simultaneously modeling time-varying tail dependence structures and heteroskedastic marginal distributions in independent but non-identically distributed random vectors. The authors propose a nonparametric method to estimate the integrated tail copula and establish its asymptotic theory. Notably, this work is the first to investigate time-varying tail dependence under heteroskedastic margins, demonstrating that heteroskedasticity does not affect the limiting distribution of the integrated tail copula estimator. This insight enables the construction of an effective statistical test for assessing whether the tail copula remains constant over time. Theoretical analysis confirms the asymptotic efficiency of the proposed estimator, and simulation studies corroborate its strong finite-sample performance and high statistical power.

This study addresses the bias inherent in tail index estimation for heavy-tailed distributions by proposing a novel estimator that integrates bias correction with empirical likelihood. The method uniquely combines bias correction techniques within an empirical likelihood framework to yield a more accurate and stable estimator, accompanied by rigorous asymptotic theory. Simulation experiments demonstrate that the proposed approach significantly outperforms existing methods in finite samples, while empirical analyses on real-world data further confirm its practical effectiveness and applicability.

Classical empirical processes fail under heavy-tailed or skewed distributions where moments (e.g., mean or variance) do not exist. To address this, we propose the trimmed functional empirical process (TFEP) framework, which relaxes the conventional finite-variance requirement. Under mild regularity conditions, TFEP converges weakly to a Gaussian process, enabling novel asymptotic theory for one-sample and two-sample hypothesis tests as well as confidence intervals. Our methodology integrates extreme-order-statistic trimming, functional empirical process analysis, and Monte Carlo simulation, supporting robust inference for trimmed means, variances, and their ratios. Empirical evaluations demonstrate that TFEP substantially outperforms classical functional empirical processes and normal approximation methods under Pareto and Cauchy distributions—yielding more accurate confidence interval coverage. We further validate its practical utility through application to real-world income data analysis.

This study addresses the challenges of modeling extremes in multivariate high-frequency financial time series—namely, cross-sectional dependence, non-stationarity, and discretization effects—by introducing a novel approach that leverages eigen-decomposition of the correlation matrix. The method projects the original series onto an orthogonal eigenbasis to disentangle market-wide, sector-specific, and idiosyncratic components. Within this decorrelated space, peak-over-threshold (POT) extreme value analysis is applied to each component separately. This work represents the first integration of eigenbasis rotation with extreme value theory in a finite-dimensional dependent system, effectively decoupling collective dynamics from individual noise. By explicitly accounting for non-stationarity and intraday seasonality, the framework enables precise quantification and attribution of tail risk arising from distinct sources.

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 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.

Existing extreme-value models struggle to flexibly characterize both marginal tail behavior and joint dependence structures of bivariate threshold exceedances under asymptotic independence. This work proposes a novel bivariate subasymptotic parametric model that, while ensuring convergence of the margins to the generalized Pareto distribution, naturally captures the evolution of extremal dependence with varying thresholds through its scale parameters. The model encompasses the standard multivariate generalized Pareto distribution as a limiting special case and accommodates a broad spectrum of tail dependence patterns. Inference is carried out via a likelihood-free neural Bayesian approach with tailored priors, enabling direct computation and interpretation of failure probabilities. Extensive simulations and an analysis of Belgian rainfall extremes demonstrate the model’s flexibility and the effectiveness of the proposed inference framework.

This work addresses the challenge of traditional extrapolation methods failing in extreme regions due to data scarcity in the tails—a common issue in machine learning. To overcome this limitation, the authors propose a unified extreme-value extrapolation framework that integrates extreme value theory with statistical learning. Built upon asymptotic representations of univariate and multivariate tail distributions, the framework combines extreme value index estimation, tail distribution modeling, and dependence structure analysis. It is applicable to both supervised and unsupervised settings and accommodates both asymptotically dependent and independent data. Empirical evaluations demonstrate that the method substantially outperforms existing approaches in tasks such as extreme quantile regression, anomaly detection, and generative AI, yielding improved accuracy and robustness in predicting rare and extreme events.