Two Sample Testing for High-dimensional Functional Data: A Multi-resolution Projection Method

It is of great interest to test the equality of the means in two samples of functional data. Past research has predominantly concentrated on low-dimensional functional data, a focus that may not hold up in high-dimensional scenarios. In this article, we propose a novel two-sample test for the mean functions of high-dimensional functional data, employing a multi-resolution projection (MRP) method. We establish the asymptotic normality of the proposed MRP test statistic and investigate its power performance when the dimension of the functional variables is high. In practice, functional data are observed only at discrete and usually asynchronous points. We further explore the influence of function reconstruction on our test statistic theoretically. Finally, we assess the finite-sample performance of our test through extensive simulation studies and demonstrate its practicality via two real data applications. Specifically, our analysis of global climate data uncovers significant differences in the functional means of climate variables in the years 2020-2069 when comparing intermediate greenhouse gas emission pathways (e.g., RCP4.5) to high greenhouse gas emission pathways (e.g., RCP8.5).