Tools for the symbolic computation of asymptotic expansions

SUMMARY This paper describes a collection of procedures for the systematic computation of asymptotic expansions that are common in statistical theory and practice: expansions of functions of sums of independent and identically distributed random variables. The procedures permit the expansion of maximum likelihood estimates, the associated deviance or drop in likelihood and more general functions of random variables with distributions involving one or more parameters. The procedures are illustrated with examples involving general and specific laws. Much of statistical theory and practice is based on asymptotic expansions. Many programs are available to assist in the numerical evaluation of such expansions, but there is a need for computational tools to assist in their derivation and symbolic evaluation. Heller (1991) shows how symbolic calculation may be used in a wide variety of statistical problems. Kendall (1988, 1990) gives procedures for the symbolic computation of expressions in the analysis of the diffusion of Euclidean shape. Silverman and Young (1987) use computer algebra to evaluate criteria on which the decision to smooth a bootstrap distribution is based. Young and Daniels (1990) apply symbolic computation to evaluate expressions in the assessment of bootstrap bias. Venables (1985) utilizes symbolic computation heavily to obtain expansions of maximum marginal likelihood estimates, most notably Fisher's A-statistic. Barndorff-Nielsen and Blasild (1986) describe procedures for the numerical calculation of Bartlett factors in cases where cumulants of the likelihood function may be specified. Most of these references involve the evaluation of complicated formulae in particular cases and not the derivation of the formulae themselves. Here we give general procedures for both the derivation of formulae and their evaluation in specific cases. The derivation of asymptotic expansions is typically a simple but laborious task. Consider, for example, the calculation of the expectation of the likelihood ratio test statistic for a one-parameter family to order 1/n. This may be accomplished in general