Two-sample testing with local community depth

In this paper, we investigate nonparametric two-sample hypothesis testing with local community depth, a measure of data depth calculated from pairwise distances between observations. Unlike many common “center-outwards” measures of data depth, local community depth captures local features of a distribution such as multimodality and can adapt to modes with different scales. This gives local community depth greater power to detect scale changes in multimodal distributions, and importantly the method is parameter-free. The local depth employed supplements existing adaptations of depth to local regions and is computationally cheaper in high dimensions than many classic depths such as halfspace depth, regression depth, simplicial depth, and Mahalanobis depth. We test hypotheses by using local community depth in the Liu–Singh test and give conditions under which our method is consistent against fixed alternatives (and has a limiting null distribution). In simulations and an application to diagnosing dengue fever, we show that local community depth outperforms other depths when the underlying distribution is multimodal or asymmetric, and is comparable to other methods when the data is unimodal and symmetric.