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Sliced inverse regression (SIR) is the most widely-used sufficient dimensionreduction method due to its simplicity, generality and computationalefficiency. However, when the distribution of the covariates deviates from themultivariate normal distribution, the estimation efficiency of SIR is ratherlow. In this paper, we propose a robust alternative to SIR - called ellipticalsliced inverse regression (ESIR) for analysing high dimensional, ellipticallydistributed data. There are wide range of applications of the ellipticallydistributed data, especially in finance and economics where the distribution ofthe data is often heavy-tailed. To tackle the heavy-tailed ellipticallydistributed covariates, we novelly utilize the multivariate Kendall's taumatrix in a framework of so-called generalized eigenvector problem forsufficient dimension reduction. Methodologically, we present a practicalalgorithm for our method. Theoretically, we investigate the asymptotic behaviorof the ESIR estimator and obtain the corresponding convergence rate under highdimensional setting. Quantities of simulation results show that ESIRsignificantly improves the estimation efficiency in heavy-tailed scenarios. Astock exchange data analysis also demonstrates the effectiveness of our method.Moreover, ESIR can be easily extended to most other sufficient dimensionreduction methods.

High Dimensional Elliptical Sliced Inverse Regression in non-Gaussian Distributions