Power transformation of the F distribution and a power normal family

To transform the F distribution to a normal distribution, two types of formula for power transformation of the F variable are introduced. One formula is an extension of the Wilson-Hilferty transformation for the chi 2 variable, and the other type is based on the median of the F distribution. Combining those two formulas, a simple formula for the median of the F distribution is derived, and its numerical accuracy is evaluated. Simplification of the formula of the Wilson-Hilferty transformation, through the median formula, leads us to construct a power normal family from the generalized F distribution. Unlike the Box-Cox power normal family, our family has a property that the covariance structure of the maximum-likelihood estimates of the parameters is invariant under a scale transformation of the response variable. Numerical examples are given to show the diff erence between two power normal families.