Multidimensional Bhattacharyya Matrices and Exponential Families

Shanbhag (1972, 1979) has characterized the distributions belonging to an exponential family on R such that the Bhattacharyya matrix is diagonal. Since then, this set of distributions has been classed by Morris (1982) and is referred to as the class of quadratic natural exponential families. In this paper we consider a multidimensional extension of Shanbhag and we obtain a characterization of the quadratic natural exponential families on Rd.