Computational aspects of likelihood-based inference for the univariate generalized hyperbolic distribution

The generalized hyperbolic distribution is among the more often adopted parametric families in a wide range of application areas, thanks to its high flexibility as the parameters vary and also to a plausible stochastic mechanism for its genesis. This high flexibility comes at some cost, however, namely the frequent difficulty of estimating its parameters due to the presence of flat areas of the log-likelihood function, so that selected points of the parameter space, while very distant, can be essentially equivalent as for data fitting. This phenomenon affects not only maximum likelihood estimation, but Bayesian methods too, since the target function is little affected by the introduction of a prior distribution. Our interest focuses in fact on maximum likelihood estimation of the Generalized hyperbolic distribution, working in the univariate case. This paper improves upon currently employed computational techniques by presenting an alternative proposal that works effectively in reaching the global maximum of the likelihood function. The paper further illustrates the above mentioned problems in a number of cases, using both simulated and real data.