The normal inverse Gaussian distribution: a versatile model for heavy-tailed stochastic processes

The normal inverse Gaussian (NIG) distribution is a recent flexible closed form distribution that may be applied as a model of heavy-tailed processes. The NIG distribution is completely specified by four real valued parameters that have natural interpretations in terms of the shape of the resulting probability density function. By choosing the parameters appropriately, one can describe a wide range of shapes of the distribution. In this paper, we discuss several of the desirable properties of the NIG distribution. In particular, we discuss the cumulant generating function and the cumulants of the NIG-variables. A particularly important property is that the NIG distribution is closed under convolution. Finally, we derive a set of very simple yet accurate estimators of the NIG parameters. Our estimators differ fundamentally from estimators suggested by other authors in that our estimators take advantage of the surprisingly simple structure of the cumulant generating function.