Shanbhag (1972
Shanbhag ( , 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 R d .
1997 Academic Press
1. Introduction
The class of quadratic natural exponential families (NEF ) on R is well known and has been described by Morris (1982) as admitting six types of distributions namely the Gaussian, Poisson, gamma, binomial, negative binomial and hyperbolic types. On R d , the full class of the quadratic NEF has not yet been completely determined. However, the subclass of simple quadratic NEF has been determined by Casalis (1996) and may be split into 2d+4 types: the d+1 Poisson Gaussian types, the d+1 negative multinomial gamma types, the multinomial and the hyperbolic types (see Section 4). Before then, Bhattacharyya (1946) has considered the covariance matrix of ( f (1) (x, m)Γf(x, m), ..., f (i) (x, m)Γf (x, m), ...) where f (x, m) is a m-parametrized density and Seth (1949) has proved that this matrix is diagonal when f is one of the six previous distributions on R. Shanbhag (1972Shanbhag ( , 1979) ) shows that the distribution assumptions are necessary as well as sufficient for the diagonality. Clearly this last result characterizes the class of the quadratic NEF. In addition, Shanbhag has established that the diagonality of the s_s (s 3) Bhattacharyya matrix is equivalent to the diagonality of the 3_3 one. The aim of the present paper is to generalize these results to R d . For that, let article no. MV971693 105 0047-259XΓ97 25.00