Local sufficiency

SUMMARY Non-Bayesian inference is considered for a scalar parameter θ based on a p-dimensional statisticXwhich need not constitute a sufficient reduction of the original data. It is shown that there exists a (p−1)-dimensional statistic A that is jointly second-order locally ancillary for θ near θ0 but that, for p > 2, A is not unique even up to nonsingular transformations. To avoid any ambiguity in the resulting inferences, we show that there exists a statistic S, to be called second-order locally sufficient, that is, to second order, independent of any second-order locally ancillary statisticA. Furthermore, S is unique up to a nonsingular transformation, is easily computed from the log likelihood ratio statistic, has a simple distribution and avoids the perplexing problem of specifying the appropriate conditioning statistic. Confidence limits based on S have the desired coverage probability conditionally on any A, and therefore unconditionally, with error O(n−1) in each tail. Finally, for vector-valued θ, we give a rather general proof of Barndorff-Nielsen's (1980) formula for the conditional distribution of θ.