Multi-sample Comparison Using Spatial Signs for Infinite Dimensional Data

Abstract: We consider an analysis of variance type problem, where the sample observations are random elements in an infinite dimensional space. This scenario covers the case, where the observations are random functions. For such a problem, we propose a test based on spatial signs. We develop an asymptotic implementation as well as a bootstrap implementation and a permutation implementation of this test and investigate their size and power properties. We compare the performance of our test with that of several mean based tests of analysis of variance for functional data studied in the literature. Interestingly, our test not only outperforms the mean based tests in several non-Gaussian models with heavy tails or skewed distributions, but in some Gaussian models also. Further, we also compare the performance of our test with the mean based tests in several models involving contaminated probability distributions. Finally, we demonstrate the performance of these tests in three real datasets: a Canadian weather dataset, a spectrometric dataset on chemical analysis of meat samples and a dataset on orthotic measurements on volunteers.