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稀疏性约束下线性泛函的最优自适应估计AI 解读

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector $\theta \in\mathbb{R}^{d}$ belongs to a class of $s$-sparse vectors with unknown $s$. We suggest an adaptive estimator achieving a nonasymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when $s$ is unknown. Furthermore, we address the issue of…

稀疏类上线性与二次泛函的极小极大估计AI 解读

For the Gaussian sequence model, we obtain nonasymptotic minimax rates of estimation of the linear, quadratic and the $\ell_{2}$-norm functionals on classes of sparse vectors and construct optimal estimators that attain these rates. The main object of interest is the class $B_{0}(s)$ of $s$-sparse vectors $\theta=(\theta_{1},\dots,\theta_{d})$, for which we also provide completely adaptive estimators (independent of $s$ and of the noise variance $\sigma $) having…