Kernel estimators of density function of directional data

Abstract

Let X be a unit vector random variable taking values on a k-dimensional sphere Ω with probability density function f( x). The problem considered is one of estimating f( x) based on n independent observation X 1,…, X n on X. The proposed estimator is of the form f n(x) = (nh k−1) −1C(h) Σ i=1 n K[ (1−x′X i) h 2 ] , x ∈ Ω, where K is a kernel function defined on R +. Conditions are imposed on K and f to prove pointwise strong consistency, uniform strong consistency, and strong L 1-norm consistency of f n as an estimator of f.