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首页> 外文期刊>Electronic Journal of Statistics >$mathbb{L}_{p}$ adaptive estimation of an anisotropic density under independence hypothesis
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$mathbb{L}_{p}$ adaptive estimation of an anisotropic density under independence hypothesis

机译：独立假设下各向异性密度的$ mathbb {L} _ {p} $自适应估计

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In this paper, we focus on the problem of a multivariate density estimation under an $mathbb{L}_{p}$-loss. We provide a data-driven selection rule from a family of kernel estimators and derive for it $mathbb{L}_{p}$-risk oracle inequalities depending on the value of $pgeq1$. The proposed estimator permits us to take into account approximation properties of the underlying density and its independence structure simultaneously. Specifically, we obtain adaptive upper bounds over a scale of anisotropic Nikolskii classes when the smoothness is also measured with the $mathbb{L}_{p}$-norm. It is important to emphasize that the adaptation to unknown independence structure of the estimated density allows us to improve significantly the accuracy of estimation (curse of dimensionality). The main technical tools used in our derivation are uniform bounds on the $mathbb{L}_{p}$-norms of empirical processes developed in Goldenshluger and Lepski [13].
机译：在本文中，我们着重于$ mathbb {L} _ {p} $-损失下的多元密度估计问题。我们提供了一系列核估计器的数据驱动选择规则，并根据$ p geq1 $的值推导出$ mathbb {L} _ {p} $风险oracle不等式。提出的估计器允许我们同时考虑基础密度及其独立性结构的近似性质。具体而言，当还使用$ mathbb {L} _ {p} $-范数测量平滑度时，我们获得了各向异性Nikolskii类规模上的自适应上限。需要强调的是，对未知密度的未知独立性结构的适应使我们能够显着提高估计的准确性（维数诅咒）。我们推导中使用的主要技术工具是Goldenshluger和Lepski [13]开发的经验过程的$ mathbb {L} _ {p} $范数的统一界线。

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