Abstract
In this chapter, the asymptotic expansion of the distribution function of the gradient test statistic is presented for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n−1/2, n being the sample size. Based on this asymptotic expansion, we perform local power comparisons among the gradient test and the competing tests (ie, the likelihood ratio [LR], Wald, and Rao score tests). The power comparisons reveal no uniform superiority property. So, the test that uses the gradient statistic, which is very simple to be computed, is as powerful as the tests that use the LR, Wald, and score statistics. The power performance of the gradient test is examined in some specific parametric models.
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