Speaker
Description
Functional Data Analysis (FDA), focusing on data composed of functions or curves, has become increasingly popular. We study reliable methods for comparing multiple groups of functional data, especially in studies involving several factors or complex designs. We introduce a new statistical approach designed for multivariate functional data. Our methods are reliable because they allow us to compare mean functions without needing strict assumptions about the error distribution (non-Gaussian errors) or requiring that the variation patterns (covariance functions) are the same across all groups (heteroscedasticity). This makes our approach broadly useful in functional multivariate analysis of variance settings. The main idea is creating tests that perform simultaneous inference for both overall group effects (global hypotheses) and specific group comparisons (local multiple hypotheses), which is essential for detailed post-hoc testing. The test statistic is determined by taking the supremum over the pointwise Hotelling's test statistic across the function domain. We show that these resulting global and multiple tests work correctly when the sample size is large (asymptotic validity). To find the critical values, we use a specific resampling technique called the parametric bootstrap, and we confirm its theoretical correctness. Extensive simulation studies confirm that our new methods work very well even with small samples. They accurately control the type I error rate and Family-Wise Error Rate (FWER) across different scenarios, often performing better than existing methods. Simulation studies also show that the supremum-based method often achieves higher power compared to other integration-based methods and other competitors. Finally, we demonstrate the practical use of our tests by analyzing a multivariate functional air pollution data set. The complete set of proposed tests is implemented in the R package gmtFD available on CRAN.
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