Statistical inference on kurtosis of elliptical distributions

TL;DR

This paper introduces a new consistent estimator for the kurtosis of elliptical distributions using U-statistics, relaxing previous assumptions and enabling confidence interval construction, validated through simulations and real data.

Contribution

It develops a novel estimation method for elliptical kurtosis that relaxes moment and dimensionality restrictions, with proven consistency and asymptotic normality.

Findings

Estimator is consistent under relaxed conditions.

Asymptotic normality allows confidence interval construction.

Validated through extensive simulations and real data analysis.

Abstract

Multivariate elliptically-contoured distributions are widely used for modeling correlated and non-Gaussian data. In this work, we study the kurtosis of the elliptical model, which is an important parameter in many statistical analysis. Based on U-statistics, we develop an estimation method. Theoretically, we show that the proposed estimator is consistent under regular conditions, especially we relax a moment condition and the restriction that the data dimension and the sample size are of the same order. Furthermore, we derive the asymptotic normality of the estimator and evaluate the asymptotic variance through several examples, which allows us to construct a confidence interval. The performance of our method is validated by extensive simulations and real data analysis.