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Local Sufficiency

Peter McCullagh
Biometrika
Vol. 71, No. 2 (Aug., 1984), pp. 233-244
Published by: Oxford University Press on behalf of Biometrika Trust
DOI: 10.2307/2336239
Stable URL: http://www.jstor.org/stable/2336239
Page Count: 12
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Local Sufficiency
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Abstract

Non-Bayesian inference is considered for a scalar parameter θ based on a p-dimensional statistic X which 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 statistic A. 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 θ̂.

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