Applications of a new power normal family

Abstract

The main purpose of this paper is to give an algorithm to attain joint normality of non-normal multivariate observations through a new power normal family introduced by the author (Isogai, 1999). The algorithm tries to transform each marginal variable simultaneously to joint normality, but due to a large number of parameters it repeats a maximization process with respect to the conditional normal density of one transformed variable given the other transformed variables. A non-normal data set is used to examine performance of the algorithm, and the degree of achievement of joint normality is evaluated by measures of multivariate skewness and kurtosis. Besides the above topic, making use of properties of our power normal family, we discuss not only a normal approximation formula of non-central F distributions in the frame of regression analysis but also some decomposition formulas of a power parameter, which appear in a Wilson–Hilferty power transformation setting.

Keywords:

• Power normal family
• non-normality
• joint normality
• measures of multivariate skewness and kurtosis