Use of approximate marginal likelihood for model selection.

Optimal invariant tests for model discrimination exist when the two models under hypotheses represent scale-regression families. These tests are based on the ratio of the marginal likelihoods of the two families, based on the maximal invariant statistics, in order to eliminate the unknown parameters from the likelihood function. However, even in cases where these functions can in principle be found, it may be difficult to make the calculations required, since the resulting formula is expressed in terms of a multidimensional integral. In this paper a simple approximation to optimal invariant tests based on the Laplace formula is discussed. The main regularity condition required is that the maximum likelihood estimates of the scale and regression parameters exist