distributional tail comparison

This paper investigates the intrinsic relationship between quantile contribution statistics and order statistics under heavy-tailed distributions. Addressing the challenge of precisely characterizing extreme-value contributions in small samples, we derive, for the first time, a closed-form expression for the joint distribution of order statistics, enabling an explicit cumulative distribution function for quantile contributions. We further establish asymptotic normality of this statistic under large-sample conditions and characterize its limiting distributional properties. Our methodology integrates order statistics theory, extreme-value analysis, asymptotic inference, and Monte Carlo simulation. Key contributions are: (1) an exact finite-sample distributional characterization of extreme-value contributions; (2) a systematic analysis of asymptotic behavior and convergence rates; and (3) a theoretically grounded, computationally tractable, and interpretable framework applicable to financial risk modeling, network anomaly detection, and biological extreme-value analysis.

Conventional p-values suffer from unintuitive interpretation, lack of comparability across test statistics, and difficulty in combining independent evidence. Method: We propose quantile-based standardized measures—s-values (measuring tail distance in semi-tail units, where each unit halves the tail probability) and ζ-values (a two-sided symmetric extension)—establishing the semi-tail unit as a unified scale of extremity. Our approach integrates quantile standardization, logarithmic transformation, Bahadur slope analysis, and additive evidence synthesis, enabling linear critical-value inference and direct summation of s-values for multi-source evidence integration. Contribution/Results: We introduce an interpretable logarithmic-scale s-value; construct the two-sided compatible ζ-value; and propose a novel asymptotic efficiency metric based on slope differences. Empirical validation across diverse settings—including standardized testing and poker hand distributions—demonstrates the naturalness and broad applicability of the framework.

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 joint identifiability and elicitability of tail risk measures—including Value-at-Risk (VaR), Expected Shortfall (ES), and Range Value-at-Risk (RVaR)—along with their associated quantiles. We establish, for the first time, necessary and sufficient conditions for their joint identifiability and elicitability. Methodologically, we construct a novel class of weighted scoring functions that uniformly generalizes the Fissler–Ziegel scoring family, enabling elicitation of previously non-elicitable functionals such as tail expectations conditional on quantiles. Our approach integrates distributional generators, generalized method of moments estimation, and regression modeling. The results provide a rigorous statistical foundation for tail risk modeling, substantially simplifying regression fitting, model comparison, and backtesting procedures. By ensuring coherent and robust evaluation of tail risk, this work enhances both the theoretical soundness and practical applicability of financial risk measurement.

This study addresses the substantial bias often introduced in meta-analyses when estimating standard deviations solely from the five-number summary—specifically, the minimum, maximum, and median—due to insufficient information, which can compromise inferential reliability. To mitigate this issue, the authors propose a novel estimation method based on a scaled Beta distribution that incorporates data shape characteristics to improve accuracy. A comprehensive sensitivity analysis is systematically conducted to quantify estimation uncertainty. Through extensive simulation studies and real-data applications, the proposed approach demonstrates markedly superior performance over conventional estimators across a variety of underlying distributions. Additionally, the authors provide an interactive web tool to facilitate practical implementation, enabling researchers to readily assess and correct potential bias in standard deviation estimates, thereby enhancing the robustness of meta-analytic findings.

Traditional efficacy metrics struggle to capture the upper-tail persistence characteristics of high responders in biosimilar assessments. This work proposes a novel quantile-based efficacy persistence function, defined as the ratio of the tail mean to the quantile function, thereby introducing the concept of expected shortfall from risk theory into clinical persistence analysis for the first time. We demonstrate its equivalence to a scaled upper-tail first-order L-moment and develop a corresponding nonparametric estimator along with a two-sample equivalence test calibrated via bootstrap inference. Simulation studies and real-data analyses show that the proposed method effectively detects upper-tail differences undetectable by median- or mean-based approaches, substantially enhancing both sensitivity and specificity in biosimilar efficacy evaluation.

Traditional simulation studies often rely on mean and standard deviation to assess the quality of asymptotic approximations; however, the existence and convergence of moments are not guaranteed by distributional convergence alone, and such approaches inadequately characterize near-normal approximations contaminated by outliers. This work proposes replacing conventional moment-based summaries with quantile-based summary statistics, specifically employing the median, median absolute deviation, and empirical confidence interval coverage as robust and universally applicable evaluation metrics. By shifting away from the mean-centered paradigm, this approach overcomes both theoretical and practical limitations inherent in moment-based methods and provides a more interpretable and reliable criterion for evaluating simulation results.

This study addresses the lack of flexible and interpretable probability distribution models suitable for upper-tail quantile analysis in clinical research. The authors propose a novel class of quantile-based effective duration functions, defined as the ratio of the mean to a given quantile, and derive a two-parameter family of non-negative distributions with closed-form expressions by incorporating Möbius transformations and natural boundary conditions. This distributional framework provides a unified characterization of tail behavior in survival data and facilitates quantile-based reliability measures and L-moment analysis. Empirical evaluation on real-world survival datasets demonstrates that the proposed method significantly outperforms existing approaches in both goodness-of-fit and model interpretability.

This study addresses the limitations of traditional causal inference methods, which primarily focus on average potential outcomes and struggle to capture treatment effects across the entire outcome distribution—particularly when outcomes are skewed or subject to detection limits. Existing quantile treatment effect estimators are often sensitive to model misspecification under such conditions. To overcome this, the authors propose two novel strategies based on a semiparametric cumulative probability model (CPM), introducing double robustness for the first time in this context. The first approach employs an inverse cumulative distribution function, while the second directly solves the efficient influence function for marginal quantiles. Both methods are extended to estimate probabilistic treatment effects and their conditional variants. The proposed estimators exhibit robustness and asymptotic normality, with simulations demonstrating strong finite-sample performance and stable variance estimation even under model misspecification. Their practical utility is further confirmed through successful application to real-world HIV data.