论文标题
概率密度规范的最小值估计：ii。速率最佳估计程序
Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures
论文作者
论文摘要
在本文中，我们制定了速率 - 在估计$ l_p $ - norm，$ p \ in（0，\ infty）$的概率密度的最佳估计过程中，来自独立观察值。假定密度是在$ r^d $，$ d \ geq 1 $上定义的，并且属于各向异性尼古尔斯基空间中的球。我们采用最小值方法和构造速率 - 在整数$ p \ geq 2 $的情况下，最佳的估计器。我们证明，取决于尼古尔斯基类的参数和Norm Index $ P $，风险渐近技术从不一致到$ \ sqrt {n} $ - 估计范围。本文的结果补充了伴侣纸\ cite {gl20}中得出的最小值下限。
In this paper we develop rate--optimal estimation procedures in the problem of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from independent observations. The density is assumed to be defined on $R^d$, $d\geq 1$ and to belong to a ball in the anisotropic Nikolskii space. We adopt the minimax approach and construct rate--optimal estimators in the case of integer $p\geq 2$. We demonstrate that, depending on parameters of Nikolskii's class and the norm index $p$, the risk asymptotics ranges from inconsistency to $\sqrt{n}$--estimation. The results in this paper complement the minimax lower bounds derived in the companion paper \cite{gl20}.
