This paper is concerned with the estimation of the partial derivatives of a probability density function of directional data on the $d$-dimensional torus within the local thresholding framework. The estimators here introduced are built by means of the toroidal needlets, a class of wavelets characterized by excellent concentration properties in both the real and the harmonic domains. In particular, we discuss the convergence rates of the $L^p$-risks for these estimators, investigating on their minimax properties and proving their optimality over a scale of Besov spaces, here taken as nonparametric regularity function spaces.

翻译：本文涉及对当地阈值框架范围内美元维值横截面方向数据概率密度函数部分衍生物的估计,此处引入的估测器是通过近似需要建造的,这是在真实和和谐领域均具有极佳浓度特性的波子类。特别是,我们讨论了这些测算器的美元-风险汇合率,调查其微量成份特性,并证明其在贝索夫空间的尺度上的最佳性,这里将它视为非对称常规功能空间。