A note on natural exponential families with cuts

Let μ be a positive measure defined on the product of two vector spaces E=E₁×E₂. Let F=F(μ) be a natural exponential family (NEF) generated by μ such that the projection of F on E₁ constitutes a NEF on E₁. This property, called a cut on E₁, has been defined and characterized by Barndorff-Nielsen (Information and Exponential Families, Wiley, Chichester) and further developed by Barndorff-Nielsen and Koudou (Theory Probab. Appl. 40 (1995) 361). Their results can be used to conclude two properties of NEFs with cuts. The first stating that a NEF F has a cut on E₁ if and only if for all random vectors (X,Y) on E₁×E₂, having a distribution in F, the regression curve of Y on X is linear. The second property states that the linearity of the scedastic curve of Y on X is a necessary condition for F to have a cut on E₁. These two properties of linearity of the regression and scedastic curves provide, in some situations, rather easily verifiable conditions for examining whether a NEF has a cut. Moreover, they are used to provide some interesting characterizations. In particular, some characterizations of the Gaussian and Poisson NEFs are obtained as special cases.