Minimax and adaptive estimation of general linear functionals under sparsity

Abstract

We study nonasymptotic minimax estimation of the linear functional $L(\theta)=\eta^\top \theta$ for a high-dimensional $s$-sparse mean vector with an arbitrary loading vector $\eta$. For symmetric noise with exponentially decaying tails, we derive the sharp minimax rate, explicit in $s$, $\eta$, the tail parameter, and the noise level. The proposed estimator combines plug-in estimation for coordinates with large loadings and thresholding for coordinates with small loadings, and the matching lower bound is obtained via a loading-dependent sparse prior. For unknown sparsity, we construct an $\eta$-dependent Lepski-type procedure and show that, for a broad verifiable class of loading vectors, its risk matches the oracle rate up to the optimal logarithmic factor. Explicit examples illustrate how heterogeneity in $\eta$ changes both the minimax and adaptive rates. We also extend the analysis to non-symmetric noise, hypothesis testing, and estimation with unknown noise variance, where we show that asymmetry can increase the minimax rate in certain examples of $\eta$. Among these results, the two main technical novelties are the following. First, we extend the sharp lower-bound theory beyond the Gaussian setting via a new $\chi^2$ bound for generalized Gaussian distributions. Second, for possibly non-symmetric noise, we derive new lower bounds through a worst-case asymmetric construction.