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Distribution-Free Pointwise Adjusted P-Values for Functional Hypotheses

Graphical tests assess whether a function of interest departs from an envelope of functions generated under a simulated null distribution. This approach originated in spatial statistics, but has recently gained some popularity in functional data analysis. Whereas such envelope tests examine deviation from a functional null distribution in an omnibus sense, in some applications we wish to do more: to obtain p-values at each point in the function domain, adjusted to control the familywise error rate. Here we derive pointwise adjusted p-values based on envelope tests, and relate these to previous approaches for functional data under distributional assumptions. We then present two alternative distribution-free p-value adjustments that offer greater power. The methods are illustrated with an analysis of age-varying sex effects on cortical thickness in the human brain.