Non-Bayesian inference is considered for a scalar unknown parameter. When the minimal sufficient statistic is of dimension greater than one and no suitable exactly ancillary statistic is available, a statistic with a certain property of local ancillarity is calculated and the conditional distribution given that statistic is evaluated by Edgeworth expansion.The resulting distribution is used to calculate significance tests and confidence intervals which have desired probability levels to O (/), where n is the sample size, and which are appropriately conditional. The resulting confidence intervals are simply related to likelihood inference in the parameterization in which Fisher's information is constant, thus generalizing Fisher's (1934) result for location parameters.
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David Cox (1980) studied this question.