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 ? 2 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 ? 2 tests are seen to exhibit very good properties in specific scenarios, but appear extremely sensitive to the shape of the alternative.