A note on the convolution of inverted-gamma distributions with applications to the Behrens-Fisher distribution

摘要

The generalized Behrens-Fisher distribution is defined as the convolution of two Student t distributions and is related to the inverted-gamma distribution by means of a representation theorem as a scale mixture of normals where the mixing distribution is a convolution of two inverted-gamma distributions. One important result in this paper establishes that for odd degrees of freedom the Behrens-Fisher distribution is distributed as a finite mixture of Student t distributions. This result follows from the main theorem concerning the form of the convolution of inverted-gamma distributions with demi-integer shape parameter.