Bias correction for kernel density estimation with spherical data

Abstract

Kernel density estimations with spherical data can flexibly estimate the shape of an underlying density, including rotationally symmetric, skewed, and multimodal distributions. Standard estimators are generally based on rotationally symmetric kernel functions such as the von Mises kernel function. Unfortunately, their mean integrated squared error does not have root-$n$ consistency and increasing the dimension slows its convergence rate. Therefore, this study aims to improve its accuracy by correcting this bias. It proposes bias correction methods by applying the generalized jackknifing method that can be generated from the von Mises kernel function. We also obtain the asymptotic mean integrated squared errors of the proposed estimators. We find that the convergence rates of the proposed estimators are higher than those of previous estimators. Further, a numerical experiment shows that the proposed estimators perform better than the von Mises kernel density estimators in finite samples in scenarios that are mixtures of von Mises densities.