We investigate the choice of default prior for use with likelihood to facilitate Bayesian and frequentist inference. Such a prior is a density or relative density that weights an observed likelihood function leading to the elimination of parameters not of interest and accordingly providing a density type assessment for a parameter of interest. For regular models with independent coordinates we develop a secondorder prior for the full parameter based on an approximate location relation from near a parameter value to near the observed data point; this derives directly from the coordinate distribution functions and is closely linked to the original Bayes approach. We then develop a modified prior that is targetted on a component parameter of interest and avoids the marginalization paradoxes of Dawid, Stone and Zidek (1973); this uses some extensions of Welch-Peers theory that modify the Jeffreys prior and builds more generally on the approximate location property. A third type of prior is then developed that targets a vector interest parameter in the presence of a vector nuisance parameter and is based more directly on the original Jeffreys approach. Examples are given to clarify the computation of the priors and the flexibility of the approach.

The paper gives an overview of modern likelihood asymptotics with emphasis on results and applicability. Only parametric inference in well-behaved models is considered and the theory discussed leads to highly accurate asymptotic tests for general smooth hypotheses. The tests are refinements of the usual asymptotic likelihood ratio tests, and for one-dimensional hypotheses the test statistic is known as r , introduced by Barndorff-Nielsen. Examples illustrate the applicability and accuracy as well as the complexity of the required computations. Modern likelihood asymptotics has developed by merging two lines of research: asymptotic ancillarity is the basis of the statistical development, and saddlepoint approximations or Laplace-type approximations have simultaneously developed as the technical foundation. The main results and techniques of these two lines will be reviewed, and a generalization to multi-dimensional tests is developed. In the final part of the paper further problems and ...

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La revue canadienne de statistique

Higher order approximations to p-values can be obtained from the log-likelihood function and a reparameterization that can be viewed as a canonical parameter in an exponential family approximation to the model. This approach clarifies the connection between Skovgaard (1996) and Fraser et al. (1999a), and shows that the Skovgaard approximation can be obtained directly using the mean log-likelihood function. Some key words: approximate pivot; Fraser information; Kullback–Leibler distance; p ∗ approximation; tangent exponential model 1.

The assessment of a vector parameter is central to statistical theory. The analysis of variance with tests and confidence regions for treatment effects is well established and the related distribution theory is conveniently quite straightforward, particularly in the normal error case. In more general contexts such as generalized linear models, the assessment is usually

Ke y Words and Phrases: asymptotic approximations, exponential model, saddlepoint approximation, tail probabilities, test quantities Saddlepoint methods, extended to distribution functions, can provide highly accurate tail probabilities for testing real parameters in exponential models. For extensions, asymptotic connections among various test quantities are needed. For five quantities, the maximum likelihood departure standardized by observed and expected information, the score function standardized by observed and expected information, and the signed square root of the likelihood ratio statistic, the needed connections to third order are recorded. Their use is illustrated by a simple integration The saddlepoint method for approximating a probability density function from a corresponding cumulant generating function was introduced to statistics by Daniels (1954) and Bardorff-Nielsen and Cox (1979). Amethod for approximating the corresponding distribution function was introduced by Lugannani and Rice (1980) and produces highly accurate tail probabilities

summary The quantity r ∗ (ψ, y) was introduced by Barndorff-Nielsen, in part as a distributional refinement of the signed likelihood root r(ψ, y), a refinement that can also be approximated by a mean and standard deviation adjustment of the root r(ψ, y). We clarify: that r ∗ is not linear in r; that r ∗ achieves large distributional improvement on r(ψ, y); and that r ∗ provides the definitive separation of inference information concerning scalar component parameters of a statistical model. These distributional and inference properties deserve broader awareness.

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Inference concerning the correlation coefficient of two random variables from the bivariate normal distribution has been investigated by many authors. In particular, Fisher [1915. Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population. Biometrika 10, 507–521] and Hotelling [1953. New light on the correlation coefficient and its transform. J. Roy. Statist. Soc. Ser. B 15, 193–232] derived various exact forms of the density for the sample correlation coefficient. However, obtaining confidence intervals based on these densities can be computational intensive. Fisher [1921. On the ‘‘probable error’ ’ of a coefficient of correlation deduced from a small sample. Metron 1, 3–32], Hotelling [1953. New light on the correlation coefficient and its transform. J. Roy. Statist. Soc. Ser. B 15, 193–232], and Ruben [1966. Some new results on the distribution of the sample correlation coefficient. J. Roy. Statist. Soc. Ser. B 28, 513–525] suggested several simple approximations for obtaining confidence intervals for the correlation coefficient. In this paper, a likelihood-based higher-order asymptotic method is proposed to obtain confidence intervals for the correlation coefficient. The proposed method is based on the results in Fraser and Reid [1995. Ancillaries and third order significance. Utilitas Math. 7, 33–53] and Fraser et al. [1999. A simple general formula for tail probabilities for frequentist and Bayesian inference. Biometrika 86, 249–264]. Simulation results indicated that the proposed method is very accurate even when the sample size is small. r 2007 Elsevier B.V. All rights reserved.