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：本文研究局部阈值框架下$d$维环面上方向数据概率密度函数偏导数的估计问题。所提出的估计量基于环面needlet构建，该类小波在实域和调和域均具有优异的集中性质。特别地，我们讨论了这些估计量的$L^p$风险收敛速率，研究了其极小极大性质，并在作为非参数正则函数空间的Besov空间尺度上证明了其最优性。