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 Sd-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 {sup(x is an element of Sd-1)vertical bar f(n)(x) - E(f(n)(x))vertical bar, n >= 1} hold.