The Power Normal distribution

Abstract

The Inverse Box-Cox (BC) transformation (see [1]) produces the Power Normal (PN) distribution family that includes the log-normal and the normal distributions, see [2], [4] and [3]. The Box-Cox power transformation aims to transform data to approximate normality (truncated normal (TN)) therefore the knowledge of the its inverse scale is of major importance. In this paper we consider the univariate PN distribution, and, because in many applications there are more than one variable and correlation we also consider the bivariate PN distribution (BPN), see [6] and [5]. We give some important results concerning both distributions. In [6] it is given a formula to approximate the ordinary moments of the PN distribution, herein, we give an exact formula for the ordinary integer moments of the PN distribution. Regarding the BPN distribution, we calculate the marginal probability density functions (pdf) and we note that they are not univariate PN distributed however the conditional pdf is PN univariate. This is also true for the corresponding functions of the transformed scale, the TN distribution. The correlation curve or curve of the conditional moments E(X 2|X 1 = x 1) that is useful to characterize the structure of the correlation is very well fitted by a power law model f(x) = axb + c. (c) Published under licence by IOP Publishing Ltd.