Kernels, Degrees of Freedom, and Power Properties of Quadratic Distance Goodness-of-Fit Tests

In this article, we study the power properties of quadratic-distance-based goodness-of-fit tests. First, we introduce the concept of a <italic>root kernel</italic> and discuss the considerations that enter the selection of this kernel. We derive an easy to use normal approximation to the power of quadratic distance goodness-of-fit tests and base the construction of a <italic>noncentrality index,</italic> an analogue of the traditional noncentrality parameter, on it. This leads to a method akin to the Neyman-Pearson lemma for constructing optimal kernels for specific alternatives. We then introduce a <italic>midpower analysis</italic> as a device for choosing optimal degrees of freedom for a family of alternatives of interest. Finally, we introduce a new diffusion kernel, called the <italic>Pearson-normal kernel</italic>, and study the extent to which the normal approximation to the power of tests based on this kernel is valid. Supplementary materials for this article are available online.