Statistica Sinica 18(2008), 299-311

IMPROVING DIMENSION REDUCTION VIA

CONTOUR-PROJECTION

Hansheng Wang$^1$, Liqiang Ni$^2$ and Chih-Ling Tsai$^{1,3}$

$^1$Peking University, $^2$University of Central Florida

and $^3$University of California-Davis

Abstract: Most sufficient dimension reduction methods hinge on the existence of finite moments of the predictor vector, a characteristic which is not necessarily warranted for every elliptically contoured distribution as commonly encountered in practice. Hence, we propose a contour-projection approach, which projects the elliptically distributed predictor vector onto a unit contour, which shares the same shape as the predictor density contour. As a result, the projected vector has finite moments of any order. Furthermore, contour-projection yields a hybrid predictor vector, which encompasses both the direction and length of the original predictor vector. Therefore, it naturally leads to a substantial improvement on many existing dimension reduction methods (e.g., sliced inverse regression and sliced average variance estimation) when the predictor vector has a heavy-tailed distribution. Numerical studies confirm our theoretical findings.

Key words and phrases: Contour-projection, dimension reduction, linearity condition, sliced average variance estimation, sliced inverse regression.