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Implicit Alternatives and the Local Power of Test Statistics

Russell Davidson
James G. MacKinnon

Abstract

The local power of test statistics is analyzed by extending the notion of Pitman sequences to sequences of data-generating process (DGP's) that approach the null hypothesis without necessarily satisfying the alternative. A space of probability densities is defined and endowed with the structure of an infinite-dimensional Hilbert manifold, which permits a geometrical interpretation of hypothesis testing. The three classical test statistics--LR, Wald, and LM--are shown to tend asymptotically to the same random variable under all sequences of local DGP's. The power of these statistics is seen to depend on the null, the alternative, and the sequence of DGP's 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" against which that statistic will have highest power, and which coincides with the explicit alternative for the classical test statistics.