  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.
