Corrections to test statistics in principal Hessian directions

By Peter M. Bentler, Jun Xie. 2000

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Dimension reduction in regression without matrix inversion (2007, methodology): “An alternative approach, see for example Bentler & Xie (2000), is to use Sattherthwaite’s chi-squared approximation.”
Dimension reduction for the conditional mean in regressions with categorical predictors (2003, methodology): “Alternatively, instead of calculating percentage points for the distribution of a combination of chi-squares, we can resort to an approach employed with satisfactory results by Bentler and Xie (2000) in the context of PHD.”
A Model-Free Test for Reduced Rank in Multivariate Regression (2003, methodology): “Alternatively, instead of calculating percentage points for the distribution of b3m, we can use Satterthwaite’s (1941) chisquared approximation, which was used with satisfactory results by Bentler and Xie (2000).”
Marginal tests with sliced average variance estimation (2007, unknown): “The p-value can be approximated by using the modified statistic 2Tn(Ĥ ) (∑ δj/∑ δ2j ), which is distributed approximately as a χ2 random variable with (s − 1) (∑ δj )2 /∑ δ2j degrees of freedom (Bentler & Xie, 2000).”
On Directional Regression for Dimension Reduction (2007, methodology): “Algorithms are available in the literature for calculating the distribution of linear combination of independent χ21 ’s, such as the foregoing (see Field 1993; Bentler and Xie 2000; Cook and Setodji 2003).”