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.
From: Jie Xie [view email]
[v1]
Mon, 29 Sep 2025 23:37:31 UTC (51 KB)
[v2]
Thu, 13 Nov 2025 08:51:44 UTC (54 KB)
[v3]
Tue, 28 Apr 2026 03:31:51 UTC (59 KB)
[v4]
Sun, 30 Aug 2026 13:49:26 UTC (64 KB)
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