What question did this study set out to answer?

This study aims to classify exponential families on the real line that remain invariant under scaling and translation.

April 16, 2026Information Geometry0 citationsOpen Access

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

KTKoichi TojoTokai University TYTaro YoshinoThe University of Tokyo

Key Points

This study aims to classify exponential families on the real line that remain invariant under scaling and translation.
Examined properties of normal distributions and exponential-polynomial distributions.
Proved closure properties of certain exponential families under scaling and translation.
Conducted mathematical evaluations to establish equivalence of families.
Demonstrated that normal distributions are closed under scaling and translation.
Established that exponential-polynomial distributions share similar closure properties.
Proved any exponential family invariant under scaling and translation is essentially exponential-polynomial.

Abstract

Abstract The family of normal distributions on R R is closed under scaling and translation. Another example of an exponential family on R R 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.

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Cite This Study

Tojo et al. (Tue,) studied this question.

synapsesocial.com/papers/69e07de52f7e8953b7cbee49 https://doi.org/https://doi.org/10.1007/s41884-026-00197-4

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