• Bernoulli
  • Volume 2, Number 2 (1996), 145-165.

An explicit large-deviation approximation to one-parameter tests

Ib M. Skovgaard

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An approximation is derived for tests of one-dimensional hypotheses in a general regular parametric model, possibly with nuisance parameters. The test statistic is most conveniently represented as a modified log-likelihood ratio statistic, just as the R*-statistic from Barndorff-Nielsen (1986). In fact, the statistic is identical to a version of R*, except that a certain approximation is used for the sample space derivatives required for the calculation of R*. With this approximation the relative error for large-deviation tail probabilities still tends uniformly to zero for curved exponential models. The rate may, however, be O(n-1/2) rather than O(n-1) as for R*. For general regular models asymptotic properties are less clear but still good compared to other general methods. The expression for the statistic is quite explicit, involving only likelihood quantities of a complexity comparable to an information matrix. A numerical example confirms the highly accurate tail probabilities. A sketch of the proof is given. This includes large parts which, despite technical differences, may be considered an overview of Barndorff-Nielsen's derivation of the formulae for p* and R*.

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Bernoulli Volume 2, Number 2 (1996), 145-165.

First available in Project Euclid: 31 October 2007

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Zentralblatt MATH identifier

conditional inference large-deviation expansions modified log-likelihood ratio test nuisance parameters parametric inference


Skovgaard, Ib M. An explicit large-deviation approximation to one-parameter tests. Bernoulli 2 (1996), no. 2, 145--165. doi:10.3150/bj/1193839221.

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