Likelihood based discrimination between separate scale and regression models.

The aim of this paper is to compare through simulation the likelihood ratio (LR) test with the most powerful invariant (MPI) test, and approximations thereof, for discriminating between two separate scale and regression models. The LR test as well as the approximate (first order) MPI test based on the leading term of the Laplace expansion for integrals are easy to compute. They only require the maximum likelihood estimates for the regression and scale parameters and the two observed informations. Even the approximate (second order) MPI test is not computationally heavy. On the contrary, the exact MPI test is expressed in terms of multidimensional integrals whose numerical evaluation appears reliable only in the two-dimensional and in the three-dimensional case. Two conclusions emerge in this paper. First, for scale and location models, exact (when computable) and approximate MPI tests are equivalent to the LR test in all the situations considered and for every sample size. This contrasts somehow with the prescription usually implied in the literature. Second, when the dimension of the regression parameter is a considerable fraction of a small or moderate sample size, the second order approximation to the MPI test clearly improves on the LR test, unlike the first order approximation.