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SHARP ERROR BOUNDS FOR ASYMPTOTIC EXPANSIONS OF

THE DISTRIBUTION FUNCTIONS FOR SCALE MIXTURES

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RYOICHI SHIMIZU^{ 1}
AND YASUNORI FUJIKOSHI^{ 2}

^{1} *The Institute of Statistical Mathematics,
4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan*

^{2} *Department of Mathematics, Hiroshima
University, Kagamiyama 1-3-1,*

Higashi-Hiroshima, Hiroshima 739, Japan
(Received January 18, 1995; revised December 26, 1995)

**Abstract.**
Let *Z* be a random variable with the
distribution function *G*(*x*) and let *s* be a positive random
variable independent of *Z*. The distribution function *F*(*x*) of the
scale mixture *X* = *sZ* is expanded around *G*(*x*) and the difference
between *F*(*x*) and its expansion is evaluated in terms of a
quantity depending only on *G* and the moments of the powers of
the variable of the form *s*^{delta/rho} -1 , where *rho*(>0) and
*delta*(= ±1) are parameters indicating the types of expansion.
For *delta* = -1 , the bound is sharp under some extra conditions.
Sharp bounds for errors of the approximations of the scale mixture
of the standard normal and some gamma distributions are given either
by analysis (*delta* = -1) or by numerical computation (*delta* = 1) .

**Key words and phrases:**
Asymptotic expansion, normal
distribution, gamma distribution, scale mixture, sharp bound.

**Source**
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