elliptical distribution theory

High-dimensional data often exhibit heavy tails, heterogeneity, and non-Gaussian characteristics, which undermine the validity of conventional covariance-based methods. This work proposes a unified robust inference framework tailored to elliptically symmetric distributions, integrating techniques such as spatial signs, spatial ranks, multivariate Kendall’s tau matrices, shape matrix estimation, and adaptive testing. By relaxing the restrictive Gaussian assumption, the framework achieves robust and efficient performance across a range of high-dimensional tasks—including location inference, covariance and precision matrix estimation, factor model testing, discriminant analysis, and dimension reduction—thereby substantially enhancing adaptability to heavy-tailed high-dimensional data.

This paper addresses the problem of testing whether a multivariate distribution is elliptical. The proposed nonparametric test is based on the Kullback–Leibler (KL) divergence, leveraging the characterization that an elliptical distribution is equivalent to independence between radial length and direction, with the latter uniformly distributed on the unit sphere. The method constructs a test statistic via k-nearest-neighbor estimation of the KL divergence. To establish first-order asymptotic normality—bypassing stringent assumptions on functional Fréchet differentiability—it employs sample splitting, truncation, and spherical projection. Furthermore, bias correction and variance inflation techniques are introduced to handle degeneracy of the influence function, enabling unified treatment under both known and unknown mean/covariance settings. Numerical experiments demonstrate that the proposed test significantly outperforms state-of-the-art methods in both size control and statistical power.

This paper addresses the limited robustness and flexibility of existing graphical model estimation methods for high-dimensional non-Gaussian data. We propose the elliptical skew SKEPTIC method, which constructs a meta-skew-elliptical copula graphical model. To our knowledge, this is the first work to extend the semiparametric elliptical distribution family to the meta-skew-elliptical class and adapt the SKEPTIC estimator to jointly accommodate skewness and heavy tails. Our approach integrates skew-elliptical distribution theory, rank-based correlation estimation, and sparse precision matrix regularization, enabling robust precision matrix estimation and graph structure recovery within a semiparametric Gaussian copula framework. We establish optimal parametric convergence rates theoretically. Simulation studies demonstrate stable graph recovery performance. Empirical analysis on S&P 500 daily log-returns shows substantial improvements in both interpretability and robustness of financial networks.

This study addresses the challenge of goodness-of-fit testing for high-dimensional elliptical models, where complex dependence between radial and directional components hinders conventional approaches. The authors propose an adaptive radial–directional dependence testing framework that first applies affine standardization to the data and then constructs test statistics based on correlations between log-radius and directional coordinates. By integrating sum-type, max-type, and Cauchy combination strategies, the method effectively detects dense, sparse, and mixed deviations from ellipticity, respectively. Theoretically, the work establishes, for the first time, the asymptotic independence of the sum and max statistics under both the null and local alternatives, and validates the efficacy of high-dimensional Hettmansperger–Randles plug-in standardization. Extensive simulations and real-data analyses demonstrate that the proposed procedure achieves accurate size control, complementary power across scenarios, and interpretable coordinate-level diagnostic capability.

This paper addresses robust Λ-quantile modeling under partial knowledge of the loss distribution, where the Λ-quantile generalizes the classical quantile via a flexible loss function Λ. To handle model uncertainty, we establish, for the first time, an equivalence between the robust Λ-quantile and the Λ-quantile under extremal distributions. Leveraging this equivalence, we derive closed-form analytical solutions for three canonical uncertainty sets: moment-constrained, Wasserstein-ball-constrained, and marginal-constrained sets. Our methodology integrates extremal distribution theory, optimal transport (Wasserstein distance), robust optimization, and risk aggregation modeling. The results unify the characterization of robustness across diverse uncertainty structures and yield computationally tractable, interpretable decision rules for optimal portfolio selection under model ambiguity.

This study addresses the suboptimal classification performance of traditional Quadratic Discriminant Analysis (QDA) under shared-generator elliptical class-conditional distributions, where the assumption of an affine radial link fails to capture the non-affine structure of the log-likelihood ratio. By analyzing the within-class radius distribution, the authors derive the Bayes-optimal radial link function and propose an asymptotically Bayes-optimal discriminant analysis method based on fractional-power random polynomial projections. They introduce, for the first time, a closed-form identifiable family of radial links that requires no spline smoothing or additional hyperparameter tuning, with theoretical guarantees of √n-consistent estimation and asymptotic normality—formally verified in Lean 4. Empirical results show competitive or superior performance against tuned generalized additive models (GAMs) on UCI and financial time-series benchmarks, notably improving accuracy on breast_cancer and heavy-tailed financial sequences (e.g., crude oil, S&P 500), while simulations confirm the √n convergence rate and vanishing excess risk.

This study addresses the failure of calibration and bias in diagonal standardization that plague high-dimensional two-sample location tests under elliptically symmetric distributions, particularly when strong covariance correlations and heavy-tailed data are present. To overcome these issues, the authors propose a novel spatial sign test based on pairwise coordinate-wise differences scaled by their marginal quantiles. The method employs a diagonal standardizer that requires no moment conditions on the radial component and accommodates arbitrary correlation structures, combined with diagonal deletion correction and a Rademacher wild bootstrap. Theoretically, the work establishes, for the first time under general dependence, a weighted chi-square null distribution and a high-dimensional stochastic expansion. Practically, it yields uniformly consistent null distribution estimation—subsuming the classical normal approximation as a special case when no dominant eigenvalues exist—and substantially enhances testing power in heavy-tailed, high-dimensional settings.

This study addresses the challenging problem of goodness-of-fit testing for high-dimensional elliptical distributions, particularly when the dimensionality is comparable to or exceeds the sample size and complex correlation structures are present. The authors propose a novel test statistic that circumvents the need to estimate the inverse of the covariance matrix, thereby avoiding instability in high dimensions. Under only finite moment conditions, they establish a Gaussian approximation for the proposed statistic and develop a theoretically justified Gaussian multiplier bootstrap procedure—the first of its kind for high-dimensional ellipticity testing. Notably, their method accommodates dimensions satisfying $\log p = o(n^{1/14})$. Numerical experiments demonstrate robust finite-sample performance and high power against various alternatives, while real-data analyses confirm its practical feasibility and effectiveness.

This study addresses the challenge of one-sample location testing in high-dimensional heavy-tailed data with strong cross-sectional dependence. The authors propose the Elliptically Regularized Hotelling Test with Cauchy Combination (ERHT–CC), which constructs a regularized test statistic based on the spatial median and sign covariance matrix, and adaptively aggregates p-values across multiple ridge parameters using the Cauchy combination method—eliminating the need to estimate cross-ridge correlations or tune hyperparameters. Theoretically, this work establishes, for the first time under elliptically symmetric heavy-tailed distributions, a regularized Hotelling test with an explicit local power function and proves its asymptotic normality under the null hypothesis. Numerical experiments demonstrate that ERHT–CC substantially outperforms existing methods in scenarios involving heavy tails and strong dependence, exhibiting excellent finite-sample performance.