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 projected GL (PGL) 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 expectation-maximization and direct maximum likelihood to assess its feasibility, while the PGL model is contrasted with the von Mises and projected normal distributions to assess its prospective utility. The results showed that quadrature-based estimation is more reliable consistently across selected scenarios and sample sizes than alternative estimation methods, while the PGL complements other distributions in terms of flexibility.