Statistical inference on kurtosis of independent component model

Abstract

Independent component model (ICM) is a widely-used population distribution in high-dimensional data analysis and random matrix theory. In this work, we study the kurtosis of the ICM, which is an important parameter in asymptotic distributions of many commonly used statistics and is also an important criterion for measuring the heavy-tailed nature of the data. Based on U-statistics, we develop an estimation method. Theoretically, we show that the proposed estimator is consistent under regular conditions, especially we relax the restriction that the data dimension and the sample size are of the same order. Furthermore, we derive the asymptotic normality of the estimator, which allows us to construct confidence intervals and hypothesis testing statistics. Computationally, we provide a fast and efficient algorithm by leveraging the matrix structure, where the computational complexity is essentially equivalent to computing the sample covariance matrix and the Gram matrix. Finally, the effectiveness of the proposed method is demonstrated through simulated data and real-world data.