High-Dimensional Mean Estimation via $\ell_1$-Penalized Normal Likelihood
Shota Katayama
Abstract
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 $\ell_{1}$ 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 $\ell_{1}$ 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.
Paper
http://dx.doi.org/10.1016/j.jmva.2014.05.005
