We develop a unified framework for $c$-sample homogeneity testing on the circle. The proposed class of $c$-sample Sobolev tests generalizes the two-sample Sobolev tests based on uniform scores and encompasses the existing multisample tests as particular cases. We further embed this class into a broader aggregation framework, where the samplewise Sobolev components are combined through general merging functions, including average-type, maximum-type, and interpolating families. Within this class, we introduce the first Anderson-Darling-type homogeneity test for circular data, and we further propose two new tests designed to detect multimodal departures from homogeneity, constructed from softmax and Poisson kernels. We derive the asymptotic null distribution of the class, prove its consistency against a broad family of fixed alternatives, and obtain the asymptotic distribution under shift-type local alternatives. The tests are distribution-free and therefore do not require resampling. A comprehensive simulation study demonstrates the strong power of the Anderson-Darling-type test compared with Cramér-von Mises-type competitors across several scenarios, and the effectiveness of the softmax and Poisson tests under several alternatives. The testing toolbox is applied to analyze the nursing patterns of polar bears.

翻译：暂无翻译