Moderate deviations and large deviations for a test of symmetry based on kernel density estimator

Let f(n) be a non-parametric kernel density estimator based on a kernel function K. and a sequence of independent and identically distributed random variables taking values in R. The goal of this article is to prove moderate deviations and large deviations for the statistic sup(x is an element of R) vertical bar f(n)(x) - f(n) (-x)vertical bar.