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<p>The family of normal distributions on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> is closed under scaling and translation. Another example of an exponential family on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> </math></EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月14日

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  • Koichi Tojo,
  • Taro Yoshino

摘要

The family of normal distributions on \(\mathbb {R}\) R is closed under scaling and translation. Another example of an exponential family on \(\mathbb {R}\) 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.