Kernel estimators of density function of directional data (Q104408)

The authors considered the estimation of a density function f(x) based on n independent observations \(X_ 1,X_ 2,...,X_ n\) on X, where X is a unit vector random variable taking values on a k-dimensional sphere \(\Omega\) with probability density f(x). The proposed estimator is of the form \[ f_ n(x)=(nh^{k-1})^{- 1}C(h)\sum K[(1-x'X_ i)/h^ 2],\quad x\in \Omega, \] where K is a kernel function defined on \(R_+\). Conditions are imposed on K and F to prove pointwise strong consistency, and strong \(L_ 1\)-norm consistency of \(f_ n\) as an estimator of f.