Asymptotic Expansions for Distributions and Quantiles with Power Series Cumulants

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.