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通过预言机方法在各向异性函数类上进行自适应估计AI 解读

We address the problem of adaptive minimax estimation in white Gaussian noise models under $\mathbb{L}_{p}$-loss, $1\leq p\leq\infty$, on the anisotropic Nikol’skii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For the proposed estimator we establish so-called $\mathbb{L}_{p}$-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence…

单指标约束回归模型中的自适应估计AI 解读

The problem of adaptive multivariate function estimation in the single-index regression model with random design and weak assumptions on the noise is investigated. A novel estimation procedure that adapts simultaneously to the unknown index vector and the smoothness of the link function by selecting from a family of specific kernel estimators is proposed. We establish a pointwise oracle inequality which, in its turn, is used to judge the quality of estimating the entire…

上确界范数损失下的多元密度估计：Oracle 方法、自适应与独立结构AI 解读

This paper deals with the density estimation on $\mathbb{R}^{d}$ under sup-norm loss. We provide a fully data-driven estimation procedure and establish for it a so-called sup-norm oracle inequality. The proposed estimator allows us to take into account not only approximation properties of the underlying density, but eventual independence structure as well. Our results contain, as a particular case, the complete solution of the bandwidth selection problem in the multivariate…

核密度估计的带宽选择：Oracle 不等式与自适应极小极大最优性AI 解读

We address the problem of density estimation with "$\mathbb{L}_{s}$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_{s}$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol’skii classes. The main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_{s}$-norms of empirical processes…