The two-dimensional hyperbolic distribution and related distributions, with an application to Johannsen's bean data

The contours of equal density of the two-dimensional hyperbolic distribution indicate that this distribution is capable of describing a very specific and simple form for departure from the two-dimensional normal distribution. One of the classical examples of two-dimensional data showing nonnormal variation is Johannsen's bean data. After a brief discussion of elementary properties of the two-dimensional hyperbolic distribution, the possibility of fitting this distribution to a set of data, obtained from Johannsen's data by a simple transformation, is considered. The two-dimensional hyperbolic distribution belongs to the class of generalized hyperbolic distributions which is shown to be closed under the formation of marginal distributions, under conditioning and under affine transformation.