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Use of goodness-of-fit procedures in high dimensional testing

It is increasingly common in functional data analysis that inference is carried out using some form of goodness-of-fit test, such as the adaptive Neyman (AN) test of Fan [Fan, J., 1996, Journal of the American Statistical Association, 91, 674–688.], or a test incorporating any of a variety of selection diagnostics such as Akaike's information criterion (AIC) and Bayesian information criterion. These procedures as well as weighted versions of the classical χ2 test are examined by power simulation in the context of functional data analysis. It is seen that the AN test exhibits good properties consistently across a range of dimensionalities and configurations of the alternative. Certain tests based on AIC are found to exhibit comparable performance at moderate dimensionalities. Weighted χ2 tests are seen to exhibit very good properties in specific scenarios, but appear extremely sensitive to the shape of the alternative.