Classification of scaling and translation invariant exponential families on the real line

Abstract The family of normal distributions on $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> is closed under scaling and translation. Another example of an exponential family on $$\mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> with this property is the family of exponential-polynomial distributions. We prove that, conversely, an exponential family with such property is essentially the family of exponential-polynomial distributions.