论文标题
概率密度规范的最小值估计：I。下限
Minimax estimation of norms of a probability density: I. Lower bounds
论文作者
论文摘要
本文处理了非参数估计$ l_p $ - norm，$ p \ in（1，\ infty）$的概率密度，$ r^d $，$ r^d $，$ d \ geq 1 $的问题。假定要估计的未知密度％属于各向异性尼古斯基空间中的球。我们采用最小值方法，并在Minimax风险方面得出了下限。特别是，我们证明估计程序的准确性实际上取决于$ p $是整数。此外，我们在估计非线性功能的问题中开发了一种通用技术，用于最小值风险的下限。所提出的技术适用于一类广泛的非线性功能，用于〜$ l_p $ - 规范估计中的下限。
The paper deals with the problem of nonparametric estimating the $L_p$--norm, $p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ from independent observations. The unknown density %to be estimated is assumed to belong to a ball in the anisotropic Nikolskii's space. We adopt the minimax approach, and derive lower bounds on the minimax risk. In particular, we demonstrate that accuracy of estimation procedures essentially depends on whether $p$ is integer or not. Moreover, we develop a general technique for derivation of lower bounds on the minimax risk in the problems of estimating nonlinear functionals. The proposed technique is applicable for a broad class of nonlinear functionals, and it is used for derivation of the lower bounds in the~$L_p$--norm estimation.
