Comparison of relative efficiency of kernel density estimator with the exponential map

This paper considers the relative efficiency of two kernel estimators which have been proposed by Park [Park, H.S. (2012). Asymptotic behavior of the kernel density estimator from a geometric viewpoint. Communications in Statistics. Theory and Methods, 41(19),3479–3496.] and Hall et al. [Hall, P., Watson, G., & Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika, 74  , 751-762.], respectively. For this, we first consider the L2L2-minimax rate of the estimator defined by the exponential map. In order to make a numerical study between these estimators, the computing flows for the relative efficiency are introduced on the 22-dimensional unit sphere. A comparison between the two estimators with two different kernels is also investigated.