A note on natural exponential families with cuts

Shaul K. Bar-Lev and Denys Pommeret

Statistics & Probability Letters, 2003, vol. 63, issue 2, 215-221

Abstract: Let [mu] be a positive measure defined on the product of two vector spaces E=E1xE2. Let F=F([mu]) be a natural exponential family (NEF) generated by [mu] 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 E1xE2, 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.

Keywords: Cut; Natural; exponential; family; Regression; curve; Scedastic; curve (search for similar items in EconPapers)
Date: 2003
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