median absolute deviation

This study addresses the limitations of classical and L-moments in characterizing distributional features under heavy-tailed or contaminated data, where they often fail to provide robust inference. The authors propose a novel robust moment system—MAD moments and MedAD moments—anchored at the median, which integrates absolute deviations, quantile slicing, and median-based expectations to establish a unified framework for robust statistical inference. These moments are well-defined for any distribution, including heavy-tailed cases lacking finite means, and possess bounded influence functions along with slice-wise robustness. Theoretical analysis and empirical experiments demonstrate that MAD moments achieve high efficiency under light-to-moderate tails, while MedAD moments remain stable even when higher-order moments do not exist. Notably, in parameter estimation for the Cauchy distribution, the proposed methods significantly outperform conventional likelihood-based approaches, offering both robustness and practical utility.

This study addresses the challenge of simultaneously achieving robustness and geometric interpretability in multivariate data analysis by proposing the Moving Median Absolute Deviation (MMAD) depth function, which introduces the median absolute deviation into a statistical depth framework for the first time. Built upon a median absolute distance functional, MMAD characterizes the arrangement of observations within the central 50% region through directional derivatives, gradient representations, and spherical boundary distributions, enabling efficient computation without complex optimization or projections. The method not only aligns with classical depth approaches in identifying central observations but also uncovers directional geometric features of the data. Furthermore, it establishes a theoretical connection to robust measures of skewness, offering both practical utility and enhanced interpretability in robust multivariate analysis.

This study addresses the lack of robustness in estimating dispersion for circular data under outlier contamination. For the first time, it extends three linear robust dispersion measures to the circular domain, analyzing their robustness through influence functions and relative deviation curves. The authors develop high-breakdown-point, high-efficiency parameter estimators tailored for von Mises and wrapped normal distributions. Furthermore, they propose a novel circular anomaly detection method and introduce circular violin plots for intuitive outlier visualization. Extensive Monte Carlo simulations and experiments on three real-world datasets demonstrate that the proposed approach significantly outperforms existing methods in both estimation accuracy and outlier detection capability.

Existing unit-interval distributions (e.g., Beta, Kumaraswamy) lack both analytical tractability and flexible skewness control for modeling positive-skewed bounded data—such as proportions or reliability metrics—on (0,1). Method: We propose the Median-Based Unit Rayleigh (MBUR) distribution, the first unit-interval distribution derived from the Rayleigh family via median parameterization. It admits closed-form probability density, cumulative distribution, and quantile functions, and its statistical properties—including moments and skewness—are rigorously characterized. Contribution/Results: Monte Carlo simulations and empirical fits demonstrate that MBUR significantly outperforms standard unit distributions under likelihood-based criteria and goodness-of-fit measures, especially in median-dominated skewed scenarios. This work bridges a theoretical gap in constructing analytically tractable unit distributions parameterized by location (median), offering a novel, interpretable tool for modeling bounded, positively skewed data.

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 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 challenge of robust parameter estimation in sinusoidal regression under heavy-tailed noise and outliers. The authors propose an efficient estimation method based on least absolute deviation (LAD), employing a coordinate descent strategy to jointly optimize amplitude and frequency parameters. The amplitude is updated via a weighted median, while the frequency is refined through a combination of periodogram-based grid search and local optimization. The algorithm’s modular design circumvents the high computational complexity of traditional simplex methods. Theoretical guarantees are established, including strong consistency and asymptotic normality of the resulting estimators. Experimental results on synthetic data and real-world time series—such as Mauna Loa CO₂ concentrations and airline passenger counts—demonstrate the method’s superior robustness and performance in the presence of non-Gaussian noise.

This study addresses the sensitivity of the conventional bivariate random-effects model to outlying studies in meta-analyses of diagnostic test accuracy, which can lead to biased estimates. To mitigate this issue, the authors propose a frequentist robust inference framework based on density power divergence, incorporating a tuning parameter that automatically down-weights anomalous studies. The method employs the Hyvärinen score for data-driven, adaptive selection of this tuning parameter, thereby enhancing robustness without requiring manual calibration. In addition to effectively identifying and attenuating the influence of outliers, the approach provides a quantitative measure of each study’s contribution to the pooled estimate, facilitating transparent sensitivity analyses. Simulation results demonstrate that, in the presence of outliers, the proposed method substantially reduces estimation bias and root mean squared error while improving confidence interval coverage, with its practical utility further illustrated in a meta-analysis of Mini-Mental State Examination (MMSE) data.

Traditional meta-analyses of median differences often exclude studies that do not report measures of dispersion such as interquartile range or range, potentially introducing selection bias. This work proposes a Direct Variance Estimation (DiVE) method that constructs a variance estimator for the pooled effect using only the reported median differences and sample sizes from each study, without requiring any dispersion statistics. The approach is grounded in asymptotic theory and validated through extensive simulations across diverse distributional settings. Results demonstrate that DiVE performs comparably to or better than conventional two-stage methods, particularly in small-sample scenarios and under various underlying distributions. By enabling inclusion of studies previously excluded due to missing dispersion information, DiVE enhances the completeness and reliability of evidence synthesis in meta-analysis.