Corrections to test statistics in principal Hessian directions

Li’s pHd method uses an asymptotic chi-squared test statistic to evaluate a hypothesized dimensionality of a reduceddimension space in a largely nonparametric setting. This statistic is based on an assumed normal distribution of the predictors. When the distributional assumption is violated, a mixture chi-squared test proposed by Cook is theoretically more appropriate. However, both tests may not perform well with small or intermediate sized nonnormal samples. We propose two corrections to Li’s statistic to enable the chi-squared approximation to be more accurate in such samples. The corrections are based on the mean and variance of the statistic of Cook’s mixture distribution. The performance of Li’s, Cook’s, and the two new statistics are compared in some small simulation studies. Results show that one of the new tests performs about as well as Cook’s, while the other performs better than the previously proposed tests. c © 2000 Elsevier Science B.V. All rights reserved