Interpolation rules suggested by asymptotic expansions

Abstract

SUMMARY We suggest a simple interpolation rule, based on Edgeworth and Cornish-Fisher expansions, which permits highly accurate numerical approximation in problems involving calculation of distributions or quantiles without requiring explicit calculation of terms in the expansions. Edgeworth expansions are often used to provide explicit corrections for asymptotic approxima- tions, as in recent work in testing and confidence procedures involving nonparametric methods (Pfanzagl, 1979; Hall, 1983; Abramovitch & Singh, 1985) and in time series, econometrics and related fields (Sargan, 1976; Phillips, 1978; Durbin, 1980; Taniguchi, 1985; Taniguchi & Maekawa, 1990). In the present note we point out a useful application of Edgeworth expansion methods in producing simple but accurate interpolation rules for deriving information from or for constructing statistical tables. Suppose the function f is tabulated for discrete values ni of its argument, and that we seek the value off for an untabulated point, say n. We suggest that, shouldf admit an Edgeworth expansion in increasing powers of n 2 or n1, then interpolation should be among values of nWf(nj) or nf(ni), respectively, rather than values of f(ni). This recommendation applies to approximations of both distribution functions and percentage points. In the latter case our reference to Edgeworth expansions should really be to Cornish-Fisher expansions. We should stress that our method does not require calculation of any terms of an Edgeworth or Cornish-Fisher expansion. It uses only existing information available in statistical tables, and knowledge that the Edgeworth expansion exists. This existence has been established for large classes of statistical problems; see, for example, Sargan (1975), Phillips (1977), Bhattacharya & Ghosh (1978), Durbin (1980) and Sargan & Satchell (1986). Aside from advantages of greater accuracy compared with traditional interpolation rules, our method allows linear and quadratic