Extending sliced inverse regression: the weighted chi-squared test

Sliced inverse regression (SIR) and an associated chi-squared test for dime nsion have been introduced as a method for reducing the dimension of regres sion problems whose predictor variables are normal. In this article the ass umptions on the predictor distribution, under which the chi-squared test wa s proved to apply, are relaxed, and the result is extended. A general weigh ted chi-squared test that does not require normal regressors for the dimens ion of a regression is given. Simulations show that the weighted chi-square d test is more reliable than the chi-squared test when the regressor distri bution digresses from normality significantly, and that it compares well wi th the chi-squared test when the regressors are normal.