Minimax estimation of a p-dimensional linear functional in sparse Gaussian models and robust estimation of the mean

Understand

We consider two problems of estimation in high-dimensional Gaussian models.

• The first problem is that of estimating a linear functional of the means of $n$ independent $p$-dimensional Gaussian vectors, under the assumption that most of these means are equal to zero.
• We show that, up to a logarithmic factor, the minimax rate of estimation in squared Euclidean norm is between $(s^2\wedge n) +sp$ and $(s^2\wedge np)+sp$.
• The estimator that attains the upper bound being computationally demanding, we investigate suitable versions of group thresholding estimators that are efficiently computable even when the dimension and the sample size are very large.