On spline approximation of sliced inverse regression

The dimension reduction is helpful and often necessary in exploring the nonparametric regression structure. In this area, Sliced inverse regression (SIR) is a promising tool to estimate the central dimension reduction (CDR) space. To estimate the kernel matrix of the SIR, we herein suggest the spline approximation using the least squares regression. The heteroscedasticity can be incorporated well by introducing an appropriate weight function. The root-n asymptotic normality can be achieved for a wide range choice of knots. This is essentially analogous to the kernel estimation. Moreover, we also propose a modified Bayes information criterion (BIC) based on the eigenvalues of the SIR matrix. This modified BIC can be applied to any form of the SIR and other related methods. The methodology and some of the practical issues are illustrated through the horse mussel data. Empirical studies evidence the performance of our proposed spline approximation by comparison of the existing estimators.