A Minimax Two-Sample Test for Functional Data via Grothendieck’s Divergence

Our proposed approach addresses the challenges associated with nonparametric two-sample testing for densely measured functional data. These challenges stem from the high dimensionality of data and the nature of the observation scheme. We introduce a novel metric concept for random functions known as Grothendieck’s divergence to overcome these challenges, which satisfies the homogeneity-zero equivalence property. Our approach uses a pre-smoothing technique on densely measured functional data and subsequently applies the empirical statistic to estimated curves. We establish the convergence rate of the test statistic without finite moment conditions and derive its asymptotic distributions under both null and alternative hypotheses. Due to the intractability of the limiting null distribution, we employ a permutation test to determine the critical value. Remarkably, our method achieves minimax optimality in the permutation test. This notable result has profound implications and validates the superiority of our proposed method. Our approach is highly versatile and suitable for handling various measurement error distributions. It also demonstrates robustness to heavy-tailed data or outliers through the ϵ-contamination model, rendering it an ideal choice for real-world scenarios. Through extensive numerical studies and real data analysis, we showcase that our method outperforms existing approaches in accuracy and robustness. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.