Large Deviations for a Test of Symmetry Based on Kernel Density Estimator of Directional Data

Assume that fis the nonparametric 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. We established that the large deviation principle for {supx∈S|f（x）-f（-x）|, n ≥ 1} holds if the kernel function is a function with bounded variation, and the density function f of the random variables is continuous and symmetric.