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Abstract
The second-order local power of a class of tests for a simple hypothesis about a multi-dimensional unknown parameter is considered. It turns out that the test procedure adjusted differently from Mukerjee (Citation1990a) has the identical second-order local power without making use of the average power criterion. The basic principle behind the power identity is that approximate third-order cumulants of the modified square-root version of the test statistic vanish. This represents a substantial extension of the second-order asymptotic results of tests in the 1980s and early 1990s.
Acknowledgments
The author would like to thank the Guest Editor Makoto Aoshima (University of Tsukuba) for affording him an opportunity to contribute to this special issue. He also thanks the Associate Editor and a referee for their valuable comments.
