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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…

高维变量选择一致性的紧条件AI 解读

We address the issue of variable selection in the regression model with very high ambient dimension, that is, when the number of variables is very large. The main focus is on the situation where the number of relevant variables, called intrinsic dimension, is much smaller than the ambient dimension $d$. Without assuming any parametric form of the underlying regression function, we get tight conditions making it possible to consistently estimate the set of relevant variables.…