A note on the local power of the LR, Wald, score and gradient tests

Abstract

This paper examines the local power of the likelihood ratio, Wald, score and gradient tests under the presence of a scalar parameter, ϕ say, that is orthogonal to the remaining parameters. We show that some of the coefficients that define the local powers remain unchanged regardless of whether ϕ is known or needs to be estimated, whereas the others can be written as the sum of two terms, the first of which being the corresponding term obtained as if ϕ were known, and the second, an additional term yielded by the fact that ϕ is unknown. The contribution of each set of parameters on the local powers of the tests can then be examined. Various implications of our main result are stated and discussed. Several examples are presented for illustrative purposes.