SHARP ERROR BOUNDS FOR ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTION FUNCTIONS FOR SCALE MIXTURES

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

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

2Department 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^{δ/ρ} − 1, where ρ(> 0) and δ(= ±1) are parameters indicating the types of expansion. For δ = −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 (δ = −1) or by numerical computation (δ = 1).

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