High-Dimensional Elliptical Sliced Inverse Regression in Non-Gaussian Distributions

Abstract

Sliced inverse regression (SIR) is the most widely used sufficient dimension reduction method due to its simplicity, generality and computational efficiency. However, when the distribution of covariates deviates from multivariate normal distribution, the estimation efficiency of SIR gets rather low, and the SIR estimator may be inconsistent and misleading, especially in the high-dimensional setting. In this article, we propose a robust alternative to SIR—called elliptical sliced inverse regression (ESIR), to analysis high-dimensional, elliptically distributed data. There are wide applications of elliptically distributed data, especially in finance and economics where the distribution of the data is often heavy-tailed. To tackle the heavy-tailed elliptically distributed covariates, we novelly use the multivariate Kendall’s tau matrix in a framework of generalized eigenvalue problem in sufficient dimension reduction. Methodologically, we present a practical algorithm for our method. Theoretically, we investigate the asymptotic behavior of the ESIR estimator under the high-dimensional setting. Extensive simulation results show ESIR significantly improves the estimation efficiency in heavy-tailed scenarios, compared with other robust SIR methods. Analysis of the Istanbul stock exchange dataset also demonstrates the effectiveness of our proposed method. Moreover, ESIR can be easily extended to other sufficient dimension reduction methods and applied to nonelliptical heavy-tailed distributions.

KEYWORDS:

• Central subspace
• Convergence rate
• Elliptical distribution
• Multivariate Kendall’s tau
• Sliced inverse regression
• Sufficient dimension reduction

Supplementary Material

ESIR-supp:

Additional proofs for Proposition 2 and Theorem 2. (pdf)

ESIR-code:

R-code for the simulation. (R File)

Acknowledgments

The authors are grateful to the editor, the associate editor and two referees for their constructive comments and suggestions which have greatly improved our article.

Additional information

Funding

Jia Zhang was supported in part by the National Natural Science Foundation of China (Grant Nos. 71991472 and 72003150), the China Postdoctoral Science Foundation, and the Joint Lab of Data Science and Business Intelligence at SWUFE. Xin Chen’s research was supported in part by startup fund at the Southern University of Science and Technology and the National Natural Science Foundation of China (Grant No. 12071205). Wang Zhou’s research was partially supported by a grant R-155-000-211-114 at the National University of Singapore.