High-dimensional mean estimation via ℓ1ℓ1 penalized normal likelihood

A new method is proposed for estimating the difference between the high-dimensional mean vectors of two multivariate normal populations with equal covariance matrix based on an ℓ1ℓ1 penalized normal likelihood. It is well known that the normal likelihood involves the covariance matrix which is usually unknown. We substitute the adaptive thresholding estimator given by Cai and Liu (2011) of the covariance matrix, and then estimate the difference between the mean vectors by maximizing the ℓ1ℓ1 penalized normal likelihood. Under the high-dimensional framework where both the sample size and the dimension tend to infinity, we show that the proposed estimator has sign recovery and also derive its mean squared error. We also compare the proposed estimator with the soft-thresholding and the adaptive soft-thresholding estimators which give simple thresholdings for the sample mean vector.