Statistics & Probability Letters 63 (2003) 215–221 A note on natural exponential families with cuts Shaul K. Bar-Lev a ; ∗ , Denys Pommeret b a Department of Statistics, University of Haifa, Mount Carmel, Haifa 31905, Israel b CREST-ENSAI, Rue Blaise Pascal, BP 37203, 35172 Bruz Cedex, France Received September 2002; received in revised form January 2003 Abstract Let  be a positive measure dened on the product of two vector spaces E = E 1 × E 2 . Let F = F ()bea natural exponential family (NEF) generated by  such that the projection of F on E 1 constitutes a NEF on E 1 . This property, called a cut on E 1 , has been dened and characterized by Barndor-Nielsen (Information and Exponential Families, Wiley, Chichester) and further developed by Barndor-Nielsen and Koudou (Theory Probab. Appl. 40 (1995) 361). Their results can be used to conclude two properties of NEFs with cuts. The rst stating that a NEF F has a cut on E 1 if and only if for all random vectors (X;Y ) on E 1 × E 2 , 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 E 1 . These two properties of linearity of the regression and scedastic curves provide, in some situations, rather easily veriable 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. c  2003 Elsevier Science B.V. All rights reserved. MSC: primary 62E10; secondary 62B99 Keywords: Cut; Natural exponential family; Regression curve; Scedastic curve 1. Introduction Let  be a positive Radon measure dened on the product of two vector spaces E = E 1 × E 2 , with d i being the dimension of E i ;i =1; 2. Let F = F () be a natural exponential family (NEF) generated by  and denote by p the projection of E on E 1 , i.e., p : E 1 × E 2 (x 1 ;x 2 ) → x 1 : * Corresponding author. E-mail address: barlev@stat.haifa.ac.il (S.K. Bar-Lev). 0167-7152/03/$-see front matter c  2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0167-7152(03)00086-5