Estimation of the generalized Laplace distribution and its projection onto the circle

The generalized Laplace (GL) distribution, which falls in the larger family of generalized hyperbolic distributions, provides a versatile model to deal with a variety of applications thanks to its shape parameters. The elliptically symmetric GL admits a polar representation that can be used to yield a circular distribution, which we call \emph{projected} GL distribution. The latter does not appear to have been considered yet in practical applications. In this article, we explore an easy-to-implement maximum likelihood estimation strategy based on Gaussian quadrature for the scale-mixture representation of the GL and its projection onto the circle. A simulation study is carried out to benchmark the fitting routine against alternative estimation methods to assess its feasibility, while the projected GL model is contrasted with other popular circular distributions. A real data example is given in Supplementary Materials.