A SCORE TEST AGAINST ONE-SIDED ALTERNATIVES

A score-type statistic, T-s, is introduced for testing H : psi = 0 aga inst K : psi greater than or equal to 0 and more general one-sided hyp otheses when nuisance parameters may be present; psi is a vector param eter. The main advantages of T-s are that it requires estimation of th e model only under the null hypothesis and that, it is asymptotically equivalent to the likelihood ratio statistic; these are precisely the reasons for the popularity of the score tests for testing against two- sided alternatives. In this sense, T-s preserves the main attractive f eatures of the classical two-sided score test. The theoretical results are presented in a general framework where the likelihood-based score function is replaced by an estimating function so that the test is ap plicable even if the exact population distribution is unknown. Computa tion of T-s is simplified by the fact that it can be computed easily o nce the corresponding two-sided statistic has been computed. The relev ance and simplicity of T-s are illustrated by discussing a data exampl e in detail. This example involves an autoregressive conditional heter oscedasticity (ARCH) model, and the objective is to test for the prese nce of ARCH effect. The null and alternative hypotheses turn out to be of the form H : psi = 0 and K : psi greater than or equal to 0 respec tively where psi is the vector of ARCH parameters. In contrast to the likelihood ratio and other equivalent forms that are currently availab le for testing against such one-sided hypotheses, we note the followin g: T-s is convenient to apply, because the full ARCH model need not be estimated subject to the inequality constraint psi greater than or eq ual to 0 and we do not need to know the exact likelihood because T-s i s based on estimating equations rather than likelihoods.