Testing of high dimensional mean vectors via approximate factor model

• We propose a two-step procedure to test the equality of two high-dimensional mean vectors.
• The dependence structure of samples is captured by an approximate factor model.
• Our procedure consists of two steps: the factor correction and the CLX test.
• The CLX test is robust to the estimation error from the factor correction.
• Our procedure allows for the time series dependence among distinct observations and the flexible dependence within an observation.

In high dimensional setting, some testing procedures of means usually require imposing sparsity conditions on the population mean vector and/or the covariance matrix underlying the observed data. However, this is rarely true in many scenarios in social science, biology, etc., where the variables are possibly highly correlated due to existence of common factors. In this paper, we assume that the correlated variables are generated from the approximate factor model. We then correct the common factors from the original data and based on the factor-corrected data we redo the test of means invented in Cai et al. (2013a,b) (CLX test for short). It turns out that, on one hand, the newly proposed testing procedure is more powerful than the CLX test based on the original data due to the increase of the signal to noise ratio, and on the other hand, we only need the sparsity condition on the covariance structure of the idiosyncratic error term which can be met more easily than that on the original data. The residual based adaptive thresholding estimator of the precision matrix of the idiosyncratic error term is proved to be accurate in terms of the L1L1 norm. Simulation studies justify our findings. A real data set is analyzed confirming the conclusions we obtained.