In this paper we describe a coherent multiple testing procedure for correlated test statistics such as are encountered in functional linear models. The procedure makes use of two different p-value combination methods: the Fisher combination method and the Šidák correction-based method. P-values for Fisher's and Šidák's test statistics are estimated through resampling to cope with the correlated tests. Building upon these two existing combination methods, we propose the smallest p-value as a new test statistic for each hypothesis. The closure principle is incorporated along with the new test statistic to obtain the overall p-value and appropriately adjust the individual p-values. Furthermore, a shortcut version for the proposed procedure is detailed, so that individual adjustments can be obtained even for a large number of tests. The motivation for developing the procedure comes from a problem of point-wise inference with smooth functional data where tests at neighboring points are related. A simulation study verifies that the methodology performs well in this setting. We illustrate the proposed method with data from a study on the aerial detection of the spectral effect of below ground carbon dioxide leakage on vegetation stress via spectral responses.
Keywords: combining correlated p-values; functional data analysis; multiple testing; permutation procedure.