卷积结构密度模型中的估计。部分II：各向异性类尺度上的自适应
Estimation in the convolution structure density model. Part II: adaptation over the scale of anisotropic classes
摘要：
本文继续了\cite{LW16}中开始的研究。在$\bR^d$上的卷积结构密度模型的框架下，我们讨论了基于$\bL_p$的自适应极小极大估计问题--各向异性Nikol'skii类尺度上的损失.我们充分刻画了极大极小风险的行为，在函数类和风险的定义中，正则性参数和范数指标之间存在不同的关系。特别是，我们证明了估计的函数的有界性导致极大极小风险的渐近性的本质改进。我们证明了在第一部分中提出的选择规则导致建设的最佳或接近最佳（对数因子）的自适应估计。
英文摘要：
This paper continues the research started in \cite{LW16}. In the framework of the convolution structure density model on $\bR^d$, we address the problem of adaptive minimax estimation with $\bL_p$--loss over the scale of anisotropic Nikol'skii classes. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. In particular, we show that the boundedness of the function to be estimated leads to an essential improvement of the asymptotic of the minimax risk. We prove that the selection rule proposed in Part I leads to the construction of an optimally or nearly optimally (up to logarithmic factor) adaptive estimator.
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