  Motivated by portfolio allocation and linear discriminant analysis, we
consider estimating a functional $\mathbf{\mu}^T \mathbf{\Sigma}^{-1}
\mathbf{\mu}$ involving both the mean vector $\mathbf{\mu}$ and covariance
matrix $\mathbf{\Sigma}$. We study the minimax estimation of the functional in
the high-dimensional setting where $\mathbf{\Sigma}^{-1} \mathbf{\mu}$ is
sparse. Akin to past works on functional estimation, we show that the optimal
rate for estimating the functional undergoes a phase transition between regular
parametric rate and some form of high-dimensional estimation rate. We further
show that the optimal rate is attained by a carefully designed plug-in
estimator based on de-biasing, while a family of naive plug-in estimators are
proved to fall short. We further generalize the estimation problem and
techniques that allow robust inputs of mean and covariance matrix estimators.
Extensive numerical experiments lend further supports to our theoretical
results.
