CONVERGENCE OF THE GRAM-CHARIER EXPANSION AFTER THE NORMALIZING BOX-COX TRANSFORMATION

RYUEI NISHII

Department of Integrated Arts and Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Naka-ku, Hiroshima 730, Japan

(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^{(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 transformation x^{(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.

Key words and phrases: Exponential dispersion model, exponential family, exponential tilting, power variance function, saddlepoint approximation, stable distribution.