Generalization for the distribution of hartley’S and nair’s Fmax statistics

Abstract

The exact distribution of Hartley’s FmaxH statistics under heterogeneity of variances with or without unequal sample sizes is given by Yanagisawa and Shirakawa (1997). The distribution of Hartley’s FmaxH statistics is further generalized to that when follows a non-central χ2 distribution instead of a central χ2 distribution. The exact distribution of Nair’s FmaxN statistics under heterogeneity of variances with or without unequal sample sizes when follows either a central χ2 or a non-central χ2 distribution is given. These exact distributions provide new statistical tests and the power of the tests under very general conditions, such as a simultaneous statistical test for the Analysis of Variance (ANOVA). For these tests, we provide a Fortran program1 which calculates, to a high accuracy, the upper tail probabilities, probability points and probability densities for FmaxH and FmaxN statistics under very general conditions. Usually the accuracy of the calculation is 13 to 14 digits for double precision arithmetic. The probability density functions under various conditions are shown and an extension statistical table for Nair’s FmaxN statistic is given when the degrees of freedom is small.