A High Dimensional Two-Sample Test Under a Low Dimensional Factor Structure

Existing high dimensional two-sample tests usually assume that different elements of a high dimensional predictor are weakly dependent. Such a condition can be violated when data follow a low dimensional latent factor structure. As a result, the recently developed two-sample testing methods are not directly applicable. To fulfill such a theoretical gap, we propose here a Factor Adjusted two-Sample Testing (FAST) procedure to accommodate the low dimensional latent factor structure. Under the null hypothesis, together with fairly weak technical conditions, we show that the proposed test statistic is asymptotically distributed as a weighted chi-square distribution with a finite number of degrees of freedom. This leads to a totally different test statistic and inference procedure, as compared with those of Bai and Saranadasa (1996) and Chen and Qin (2010). Simulation studies are carried out to examine its finite sample performance. A real example on China stock market is analyzed for illustration purpose.