Large deviations and moderate deviations for kernel density estimators of directional data

Let f n be the non-parametric kernel density estimator of directional data based on a kernel function K and a sequence of independent and identically distributed random variables taking values in d-dimensional unit sphere S d.1. It is proved that if the kernel function is a function with bounded variation and the density function f of the random variables is continuous, then large deviation principle and moderate deviation principle for {supx.Sd.1|fn(x).E(fn(x))|,n.1} hold.