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CONVERGENCE OF THE GRAM-CHARIER EXPANSION

AFTER THE NORMALIZING BOX-COX TRANSFORMATION

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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^{(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.

**Source**
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