Two-sample testing in high dimensions

We propose new methodology for two-sample testing in high dimensional models. The methodology provides a high dimensional analogue to the classical likelihood ratio test and is applicable to essentially any model class where sparse estimation is feasible. Sparse structure is used in the construction of the test statistic. In the general case, testing then involves non-nested model comparison, and we provide asymptotic results for the high dimensional setting. We put forward computationally efficient procedures based on data splitting, including a variant of the permutation test that exploits sparse structure. We illustrate the general approach in two-sample comparisons of high dimensional regression models (differential regression') and graphical models (differential network'), showing results on simulated data as well as data from two recent cancer studies.