Abstract: In sufficient dimension reduction, the second-order inverse regression methods, such as the principal Hessian directions and directional regression, commonly require the predictor to be normally distributed. In this paper, we introduce a type of elliptical distributions called the quadratic variance ellipticity family, which covers and approximates a variety of commonly seen elliptical distributions, with the normal distribution as a special case. When the predictor belongs to this family, we study the properties of the second-order inverse regression methods and adjust them accordingly to preserve consistency. When the dimension of the predictor is sufficiently large, we show the consistency of the conventional methods, which strengthens a previous result in Li and Wang (2007). Simulation studies and data analysis are conducted to illustrate the effectiveness of the adjusted methods.

Key words and phrases: Central mean subspace, central subspace, directional regression, principal Hessian directions, quadratic variance ellipticity family.