Likelihood-based discrimination between separate scale and regression models

The aim of this paper is to compare, in terms of power 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 when the dimension of the regression parameter is very low. Several error distributions have been considered in the simulation study, including the normal, Cauchy, skew-normal, Student's t, logistic and extreme value distributions. Two conclusions emerge in this paper. First, for scale and location models, corresponding to a constant regression model, exact (when computable) and approximate MPI tests do not improve on 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.