Convergence of the Gram-Charier expansion after the normalizing Box-Cox transformation

Abstract

Consider an exponential family such that the variance function is given by the power of the mean function. This family is denoted by ED^α if the variance function is given by μ^{(2-α)/(1-α)}, where μ is the mean function. When 0≤α<1, it is known that the transformation of ED^(α) to normality is given by the power transformationX ^{(1-2α)/(3-3α)}, and conversely, the power transformation characterizes ED^(α). Our principal concern will be to show that this power transformation has an another merit, i.e., the density of the transformed variate has an absolutely convergent Gram-Charier expansion.