A NewFApproximation for the Pillai—Bartlett Trace UnderH₀

Pillai suggested two approximations for the Pillai-Bartlett trace statistic in the null case. The first one matches one moment of a β1 random variable, and corresponds to an F random variable, and the second matches four moments in the Pearson system. Although intuitively appealing and widely used in current statistical packages, the first lacks accuracy even with moderate sample size. The second matches two moment ratios in the Pearson system and provides much greater accuracy. Two new approximations match two moments of a β1 random variable, and hence correspond to an F random variable, yet achieve most of the accuracy of Pillai's second approximation. The second of the two new approximations provides the best combination of logical properties and numerical accuracy.