Accurate confidence intervals when nuisance parameters are present

Let be an estimate of an unknown parameter ω in R pwith distribution determined by ω. (It is assumed that is not a function of a lattice random variable (r.v.).) Suppose that for r ≥ 1 the rth order cross-cumulants of have magnitude n 1-r and can be expanded in powers of n-1 where n is known. For example may be a function of the mean of n identically distributed random vectors. Let t(ω) be a real function on Rpwith finite derivatives. We show how to obtain confidence intervals (C.I.s) for t(ω) of level for any given jWe call this a jth order C.I.. The expansion for the confidence limit is generally divergent so that it makes no sense to increase j indefinitely. Thus only a limited (though possibly large) number of significant digits are available for the confidence limit - a sort of ‘statistical uncertainty principle’. For n less than a certain limit (which we give), only the usual first order C.I. is available - that based on the approximate normality of the Studentised statistic. In many cases higher order C.I.s are available even for very small sample sizes. When no parameters are present these results include the well known expansions of Cornish and Fisher(1937) and Fisher and Cornish(1960) - such as the percentiles of the Chi-square, Student and Z (equivalent to the F) distributions: as these expansions diverge, the number of significant digits available increases with the degrees of freedoms. Other well known special cases are the solutions to the Behrens-Fisher problem and the problem of obtaining a C.I. for a variance component given by Welch(1947, 1965). Over thirteen examples are given including C.I.s for interaction variances and intra-class correlation coefficients in a k-way random effects model, and a C.I. for the non-centrality parameter.