Abstract: Two-sample inference for population mean functions is a fundamental problem in functional data analysis. In recent years, projection-based testing has gained popularity, which constructs a test statistic by projecting functional observations into a finite-dimensional space. However, the criterion for selecting projection functions remains an open question, given the various types of functional spaces. In this paper, we introduce a novel measure of information loss caused by projection and provide the first theoretical analysis of the relationship between testing efficiency and the selection of projection functions. This analysis contributes to the understanding of projection-based testing and provides guidelines for selecting projection functions. Specifically, we derive the theoretical optimal projective space that achieves the best power and investigate three practical projective spaces. Tests based on these three projective spaces exhibit superior performance in both simulations and real data.

Key words and phrases: Information loss, optimal projection function, projection-based testing, selection of projection function, two-sample test.