A Score Test against One-Sided Alternatives

A score-type statistic, T s , is introduced for testing H: ψ = 0 against K: ψ ≥ 0 and more general one-sided hypotheses when nuisance parameters may be present; ψ is a vector parameter. The main advantages of T s , are that it requires estimation of the 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 features 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 applicable even if the exact population distribution is unknown. Computation of T s , is simplified by the fact that it can be computed easily once the corresponding two-sided statistic has been computed. The relevance and simplicity of T s are illustrated by discussing a data example in detail. This example involves an autoregressive conditional heteroscedasticity (ARCH) model, and the objective is to test for the presence of ARCH effect. The null and alternative hypotheses turn out to be of the form H: ψ = 0 and K: ψ ≥ 0 respectively where ψ is the vector of ARCH parameters. In contrast to the likelihood ratio and other equivalent forms that are currently available for testing against such one-sided hypotheses, we note the following: T s is convenient to apply, because the full ARCH model need not be estimated subject to the inequality constraint ψ ≥ 0 and we do not need to know the exact likelihood because T s is based on estimating equations rather than likelihoods.