Canonical exponential family parameters

Inference about a parameter θ based on the likelihood function is often based on first order approximations to standard summary statistics from the likelihood, such as the likelihood ratio statistic or the standardized maximum likelihood estimate. For more accurate inference refinements are needed to both improve the first order approximation of the distribution of the relevant statistic, and to properly take account of nuisance parameters, in the multi-parameter setting. A great many examples are available that illustrate the failure of first-order methods in models with large numbers of nuisance parameters, and there is a similar wealth of examples illustrating the accuracy of so-called higher order approximations. Recent book-length treatments include Barndorff-Nielsen and Cox (1994), Severini (2000) and Brazzale et al. (2007). In this note we consider the role of an approximating exponential family model, and in particular its canonical parameter, in obtaining higher order approximations. Connections are made between the approximations due to Barndorff-Nielsen (1986), Skovgaard (1996), Severini (1998) and Fraser et al. (1999a), and these connections are used to suggest a new method for computation of higher order approximations. ∗Department of Statistics, University of Toronto, 100 St. George St., Toronto, M5S 3G3, Canada. [email protected], [email protected]