**Abstract:** | * We propose an inferential procedure for functional data, able to select the intervals of the domain imputable of rejecting a functional null hypothesis.
The procedure is based on three different steps: (i) a functional test is performed on any interval of the domain; (ii) an unadjusted and an adjusted p-value function are defined from the results of the previous tests; (iii) the significant intervals of the domain are selected by thresholding the unadjusted or the adjusted p-value functions, depending on the desired type of control of the family-wise error rate (i.e., point-wise or interval-wise, respectively). In detail, we prove that the newly defined unadjusted p-value function provides a control of the point-wise error rate (i.e., given any point of the domain where the null hypothesis is not violated - in an L2 sense to be suitably defined - the probability of wrongly selecting it as significant is controlled) and that it is point-wise consistent (i.e., given any point of the domain where the null hypothesis is violated - in an L2 sense to be suitably defined - the probability of selecting it as significant goes to one as the sample size goes to infinity). Similarly, we prove that the newly defined adjusted p-value function provides instead a control of the interval-wise error rate (i.e., given any interval of the domain where the null hypothesis is almost-everywhere not violated the probability of wrongly selecting it as significant is controlled) and that it is interval-wise consistent (i.e., given any interval of the domain where the null hypothesis is almost-everywhere violated the probability of selecting it as significant goes to one as the sample size goes to infinity).
The procedure is also applied - together to other two state-of-the-art procedures - to the analysis of of the Canadian daily temperatures, to test for pairwise differences between four climatic regions. In detail, we show how the new procedure hereby proposed is able to give a new deeper and useful insight on the possible rejection of the null hypothesis that consists in the selection of the periods of the years presenting significant differences between each couple of regions.* |