Dimension choice for Sliced Inverse Regression based on ranks

Sliced Inverse Regression is a method for reducing the dimensionality in multivariate non parametric regression problems. While the selection of the dimensionality has been investigated for the original version, no solution has been proposed for Hsing and Carroll (1992) approach based on order statistics and associated concomitant variables. By using model selection approaches, we propose here two ways for selecting the dimensionality by estimating a loss function: first, a direct estimation is proposed and, then a Jack-Knifed estimate is investigated. Finally, the rank version is compared to classical SIR on a real life data set. 1 Introduction Consider a random variable (X; Y ) with X in ! p and Y in ! and let (X i ; Y i ) be n independent variables with same distribution as (X; Y ): To simplify, we assume that X is normally distributed. Let r be the regression function of Y in X. Then, we have Y = r(X) + ". To overcome the problem of the curse of the dimensionality, Sliced Inver..