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 $ \left\{ {\sup _{x \in S^{d - 1} } |f_n (x) - E(f_n (x))|,n \geqslant 1} \right\} $ \left\{ {\sup _{x \in S^{d - 1} } |f_n (x) - E(f_n (x))|,n \geqslant 1} \right\} hold.