Local likelihood estimation for multivariate directional data.

Our aim is to extend local likelihood methodology to circular density estimation. The idea lies in optimizing a spatially weighted version of the log-likelihood function, where the logarithm of the density is approximated by a polynomial. Advantages of such an approach would amount to more flexibility near the boundary and bias reduction when the polynomial degree increases, especially for heavy tailed distributions (as it is often the case for directional models) in higher dimensions. The use of d-fold products (d ≥ 1) of von Mises densities as weight functions facilitates the computational burden, specifically it makes possible to avoid numerical integration by exploiting the properties of Bessel functions. Our findings consist of theoretical reasoning along with simulation experiments.