Implicit Alternatives and the Local Power of Test Statistics

The local power of test statistics is analyzed by extending the notion of Pitman sequences to sequences of data-generating processes (DGPs) that approach the null hypothesis without necessarily satisfying the alternative hypothesis. Under quite general conditions, the three classical test statistics -- likelihood ratio, Wald, and Lagrange multiplier -- are shown to tend asymptotically to the same random variable under all sequences of local DGPs. The power of these tests depends on the null, the alternative, and the sequence of DGPs, in a simple and geometrically intuitive way. Moreover, for any test statistic that is asymptotically Chi-squared under the null, there exists an "implicit alternative hypothesis" which coincides with the explicit alternative for the classical test statistics, and against which the test statistic will have highest power.