Combining dependent tests based on data depth with applications to the two-sample problem for data of arbitrary types

Combining multiple tests has many real world applications. However, most existing methods fail to directly take into account the underlying dependency among the tests. In this paper, we propose a novel procedure to combine dependent tests based on the notion of data depth. The proposed method can automatically incorporate the underlying dependency among the tests, and is nonparametric and completely data-driven. To demonstrate its application, we apply the proposed combining method to develop a new two-sample test for data of arbitrary types when the data can be metrizable and their information can be characterised by interpoint distances. Our simulation studies and real data analysis show that the proposed test based on the new combining method performs well across a broad range of settings and compares favourably with existing tests.