Central limit theorem for integrated square error of kernel estimators of spherical density

Let X,,,X, be lid observations of a random variable X with probability density function f(x) on the q-dimensional unit sphere Ohm (q), in Rq+1, q greater than or equal to 1. Let f(n)(x) = n(-1)c(h) Sigma K-n(i=1)[(1-x'X-i)/h(2)] be a kernel estimator of f(x). In this paper we establish a central limit theorem for integrated square error of f(n) under some mild conditions.