Reversed Score and Likelihood Ratio Tests

Two extensions of a model in the presence of an alternative model are proposed. The extensions are based on the score function of the alternative model. It is shown that the encompassing hypothesis is equivalent to standard conditions on the score of each of the extended models. The condition on the first extension gives rise to the standard score encompassing test, while the condition on the second extension induces a so-called reversed score encompassing test. A similar logic is applied to the likelihood ratio, thus generating a likelihood ratio and a reversed likelihood ratio encompassing test. The ensued test statistics can be based on simulations if certain calculations are to difficult to carry out analytically. We study the first-order asymptotic properties of the proposed test statistics under general conditions.