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 X1…, Xn on X. The proposed estimator is of the form fn(x)=(nhk-1)-1C(h) Σni=1K[(1-x′Xi)/h2], xɛΩ, 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 L1-norm Consistency Of fn as an estimator Of f.