On approximating probability distributions of a statistic of brown-forsythe from normal and non-normal universes

Abstract

This article considers a one-way ANOVA model as given in (1.1) with heterogeneous variances. The distributions of the Brown-Forsythe F ^* statistic for this model are derived and approximated by Laguerre polynomial expansions. The Type 1 error and power function values are then evaluated for both normal and mixtures of two normals populations. For the example with k=3n ₁=8,n ₂=10 and n ₃=12. it is shown that the Satterthwaite approximation provides a close approximation to the null and non-null distributions in most o\ the cases. It is shown that the F ^* is quite robust with respect to departure from normality for Type 1 error in the case of mixtures of two normals.

Keywords:

• Departure from normality
• F ^*test
• Laguerre polynomial expansion
• mixtures of two normals
• robustness
• Satterthwaite approximation