Local Inference for Functional Data Controlling the Functional False Discovery Rate

A topic which is becoming more and more popular in Functional Data Analysis is local inference, i.e., the continuous statistical testing of a null hypothesis along a domain of interest. The principal issue in this topic is the infinite amount of tested hypotheses, which can be seen as an extreme case of multiple comparisons problem. A number of quantities have been introduced in the literature of multivariate analysis in relation to the multiple comparisons problem. Arguably the most popular one is the False Discovery Rate (FDR), that measures the expected proportion of false discoveries (rejected null hypotheses) among all discoveries. We define FDR in R, the setting of functional data defined on a compact set of R. A continuous version of the Benjamini-Hochberg procedure is introduced, along with a definition of adjusted -value function. Some general conditions are stated, under which the functional Benjamini-Hochberg (fBH) procedure provides control of FDR. We show how the procedure can be plugged-in with every parametric or nonparametric pointwise test, given that such test is exact. Finally, the proposed method - together with a nonparametric test - is applied to the analysis of the benchmark dataset of Canadian temperatures.