  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.
