Generalization for the distribution of Hartley's and Nair's Fmax statistics
The exact distribution of Hartley's Fmax(H) statistics under heterogeneity of variances with or without unequal sample sizes is given by Yanagisawa and Shirakawa (1997). The distribution of Hartley's Fmax(H) statistics is further generalized to that when nuS(2)/sigma (2) follows a non-central chi (2) distribution instead of a central chi (2) distribution. The exact distribution of Nair's Fmax(N) statistics under heterogeneity of variances with or without unequal sample sizes when nuS(2)/sigma (2) follows either a central chi (2) or a non-central chi (2) 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 program(1) which calculates, to a high accuracy, the upper tail probabilities, probability points and probability densities for Fmax(H) and Fmax(N) 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 Fmax(N) statistic is given when the degrees of freedom is small.
hartley's fmax(h) statistic heterogeneity of variances nair's fmax(n) statistic non-central chi(2) distribution