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SummaryLet ~ be an estimate of a parameter co in R p, n a known real parameter, and t(.) a real function on R p. Suppose that the variance of nln (t(&:)--t(co)) tends to a2>0 as n--,co, and that D: is an estimate of a. We give asymptotic expansions for the distributions and quantiles of Y,,~ = n~na-~(t(~,,)-t(~o))and Y~2 = n~/~;'(t(3~)-t(co))to within O(n-si~'). It is assumed that (i) E&~--~oJ as n--.co ; (ii) t(.) is suitably differentiable at w; (iii) for r>__l the rth order cross-cumulants of $~ have magnitude n ~-~ as n-~oo and can be expanded as a power series in n-~; (iv) that &~ has a valid Edgeworth expansion. (Bhattacharya and Ghosh [1] have given easily verifiable sufficient conditions for commonly used statistics like functions of sample moments and the m.l.e.)As an application we investigate for what parameter ranges common confidence intervals for a linear combination of the means of normal samples are adequate.

SUMMARY Cornish and Fisher (1937) gave the first few terms of the formal expansions for the distribution and percentiles of an asymptotically normal random variable (r.v.) in terms of its cumulants. They also showed that these expansions can sometimes produce extremely accurate approximations. In practice the cumulants are first expanded as a power series in a known parameter n (such as the sample size or degrees of freedom), and these expansions are substituted into the Edgeworth and Cornish‐Fisher expansions, which are then rearranged in powers of 1/√n and truncated. This paper gives explicit formulae for the general terms both for the Cornish‐Fisher expansions and for the derived expansions.