Estimation of Densities and Derivatives of Densities with Directional Data

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Local polynomial regression for circular predictors (2009, background): “Our kernels include kernels on the sphere which are functions ofκ{1− cos(θ)} studied by Beran (1979), Hall et al. (1987), Bai et al. (1988)and Klemelä (2000).”
ANALYZING THE RANDOM COEFFICIENT MODEL NONPARAMETRICALLY (2009, unknown): “…for spherical data is given by f̂ S(s) = c(g) n n ∑ i=1 G ( g−2 ( 1− sT Si )) , s ∈ Sd−1, (9) where G : [0,∞) → R is a kernel function, g > 0 is the smoothing parameter, and c(g) is the normalization constant: c(g)−1 = ∫ Sd−1 G ( g−2 ( 1− sT e))dμ(s), for any e ∈ Sd−1, see, e.g., Klemelä (2000).”
Exact risk improvement of bandwidth selectors for kernel density estimation with directional data (2013, background): “Some of the results by [18] were extended by [25], who studied the estimation of the Laplacian of the density and other types of derivatives.”