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Testing in functional data analysis using quadratic forms

Tests of hypotheses associated with the functional linear model are investigated under smoothness assumptions. The tests considered are those which use a quadratic-form test statistic calculated on a high-dimensional discrete model that is obtained by Fourier transformation. Asymptotic performance bounds for this class of tests are deduced under rates-of-testing theory, and explicit formulas are given that characterize the performance of many such tests. Examples are discussed, including an optimal class of tests based on quadratic forms, and recommendations are made for the use of the tests in practice. Among other insights, results describe the impact of model dimension on performance, which is a particular concern in functional data analysis.