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Tail probability approximations. (English) Zbl 0614.62016

This paper is a very interesting and clear review on two explicit approximations for the tail probability of a sample mean. The Edgeworth type expansion gives an asymptotic expansion in powers of \(n^{-1/2}\) when \(\bar x-E(X)=O(n^{-1/2})\). For large deviations it can be further reduced to an expansion whose terms decrease by a factor \(n^{-1}.\)
A recent approach, due to R. Lugannani and S. Rice [Adv. Appl. Probab. 12, 475-490 (1980; Zbl 0425.60042)] is based on an extension of the saddlepoint technique. It yields an asymptotic expansion whose terms decrease in order by a factor \(n^{-1}\) over the whole range of \(\bar x.\) The novelty of the paper is that it covers the case of lattice variables.
Reviewer: J.G.Szekely

MSC:

62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 0425.60042
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