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

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.