(Received July 19, 1991; revised April 17, 1992)
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(alpha) if the variance function is given by mu(2-alpha)/(1-alpha), where mu is the mean function. When 0 < alpha < 1, it is known that the transformation of ED(alpha) to normality is given by the power transformation x(1-2alpha)/(3-3alpha), and conversely, the power transformation characterizes ED(alpha). 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.
Key words and phrases: Exponential dispersion model, exponential family, exponential tilting, power variance function, saddlepoint approximation, stable distribution.