  In this paper, we present several estimators of the diagonal elements of the
inverse of the covariance matrix, called precision matrix, of a sample of iid
random vectors. The focus is on high dimensional vectors having a sparse
precision matrix. It is now well understood that when the underlying
distribution is Gaussian, the columns of the precision matrix can be estimated
independently form one another by solving linear regression problems under
sparsity constraints. This approach leads to a computationally efficient
strategy for estimating the precision matrix that starts by estimating the
regression vectors, then estimates the diagonal entries of the precision matrix
and, in a final step, combines these estimators for getting estimators of the
off-diagonal entries. While the step of estimating the regression vector has
been intensively studied over the past decade, the problem of deriving
statistically accurate estimators of the diagonal entries has received much
less attention. The goal of the present paper is to fill this gap by presenting
four estimators---that seem the most natural ones---of the diagonal entries of
the precision matrix and then performing a comprehensive empirical evaluation
of these estimators. The estimators under consideration are the residual
variance, the relaxed maximum likelihood, the symmetry-enforced maximum
likelihood and the penalized maximum likelihood. We show, both theoretically
and empirically, that when the aforementioned regression vectors are estimated
without error, the symmetry-enforced maximum likelihood estimator has the
smallest estimation error. However, in a more realistic setting when the
regression vector is estimated by a sparsity-favoring computationally efficient
method, the qualities of the estimators become relatively comparable with a
slight advantage for the residual variance estimator.
