Inverse Box–Cox: The power-normal distribution

Box–Cox transformation system produces the power normal (PN) family, whose members include normal and lognormal distributions. We study the moments of PN and obtain expressions for its mean and variance. The quantile functions and a quantile measure of skewness are discussed to show that the PN family is ordered with respect to the transformation parameter. Chebyshev–Hermite polynomials are used to show that the correlation coefficient is smaller in the PN scale than the original scale. We use the Fréchet bounds to obtain expressions for the lower and upper bounds of the correlation coefficient. A numerical routine is used to compute the bounds. The transformation parameter of the PN family is used to investigate the effects of model uncertainty on the upper quantile estimates.