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.

翻译：以组合分配和线性差异分析为动力,我们考虑估算一个功能值$mathbf=mu{T\mathbf=Sigma}-1}\mathbbf=1}\mathbf=mu}美元,其中平均矢量为$mathbf=Sigma}美元,常量基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数基数。