High Dimensional Elliptical Sliced Inverse Regression in non-Gaussian Distributions

TL;DR

This paper introduces elliptical sliced inverse regression (ESIR), a robust method for high-dimensional, heavy-tailed elliptically distributed data, improving estimation efficiency over traditional SIR especially in finance and economics.

Contribution

The paper develops a novel ESIR method utilizing the multivariate Kendall's tau matrix for robust dimension reduction in heavy-tailed distributions, with theoretical and practical validation.

Findings

ESIR outperforms SIR in heavy-tailed scenarios.

Simulation results confirm improved estimation efficiency.

Real data analysis demonstrates ESIR's effectiveness.

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 the covariates deviates from the multivariate normal distribution, the estimation efficiency of SIR is rather low. In this paper, we propose a robust alternative to SIR - called elliptical sliced inverse regression (ESIR) for analysing high dimensional, elliptically distributed data. There are wide range of applications of the 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 utilize the multivariate Kendall's tau matrix in a framework of so-called generalized eigenvector problem for sufficient dimension reduction. Methodologically, we present a practical algorithm for our method. Theoretically, we investigate the asymptotic behavior of the ESIR estimator and obtain the corresponding convergence rate under high dimensional setting. Quantities of simulation results show that ESIR significantly improves the estimation efficiency in heavy-tailed scenarios. A stock exchange data analysis also demonstrates the effectiveness of our method. Moreover, ESIR can be easily extended to most other sufficient dimension reduction methods.