## Bernoulli

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

### An explicit large-deviation approximation to one-parameter tests

#### Abstract

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*^{*}.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 31 October 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.bj/1193839221

**Digital Object Identifier**

doi:10.3150/bj/1193839221

**Mathematical Reviews number (MathSciNet)**

MR1410135

**Zentralblatt MATH identifier**

1066.62508

**Keywords**

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

#### Citation

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