Annals of the Institute of Statistical Mathematics

, Volume 39, Issue 3, pp 611–622

# Error bounds for asymptotic expansion of the scale mixtures of the normal distribution

• Ryoichi Shimizu
Article

## Summary

LetX be a standard normal random variable and let σ be a positive random variable independent ofX. The distribution of η=σX is expanded around that ofN(0, 1) and its error bounds are obtained. Bounds are given in terms of E(σ2V−σ2−1)k whereσ2Vσ−2 denotes the maximum of the two quantitiesσ2 andσ−2, andk is a positive integer, and of E(σ2−1)k, ifk is even.

## Key words and phrases

Normal distribution t-distribution asymptotic expansion error bound

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