A new ranking scheme for modern data and its application to two-sample hypothesis testing

Rank-based approaches are among the most popular nonparametric methods for univariate data in tackling statistical problems such as hypothesis testing due to their robustness and effectiveness. However, they are unsatisfactory for more complex data. In the era of big data, high-dimensional and non-Euclidean data, such as networks and images, are ubiquitous and pose challenges for statistical analysis. Existing multivariate ranks such as component-wise ranks, spatial ranks, and depth-based ranks do not apply to non-Euclidean data and have limited performance for high-dimensional data. Instead of dealing with the ranks of observations, we propose two types of ranks applicable to complex data based on a similarity graph constructed on observations: a graph-induced rank defined by the inductive nature of the graph and an overall rank defined by the weight of edges in the graph. To illustrate their utilization, both the new ranks are used to construct test statistics for the two-sample hypothesis testing, which converge to the $\chi_2^2$ distribution under the permutation null distribution and some mild conditions of the ranks, enabling an easy type-I error control. Simulation studies show that the new method exhibits good power under a wide range of alternatives compared to existing methods. The new test is illustrated on the New York City taxi data for comparing travel patterns in consecutive months and a brain network dataset comparing male and female subjects.