Bartlett's, Cochran's, and Hartley's Tests on Variances are Liberal When the Underlying Distribution is Long-Tailed

Abstract

For samples of equal size, the p values of Bartlett's, Cochran's, and Hartley's tests for the equality of several variances, obtained under the normality assumption, are shown to underestimate the true p values when the parent distribution is a normal or a X ² ₁ scale mixture or when there is within-sample dependence. The proof uses elements of majorization theory. When the sample sizes are not equal, a partial extension is obtained for the two-sample case.

Key Words:

• Majorization
• Normal mixture
• Schur convexity
• Star-shaped ordering
• X ² distribution