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<p>In this paper, we consider strongly unimodal distribution functions <i>V</i> whose supports lie in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {R}_+ \equiv [0, \infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">R</mi> <mo>+</mo> </msub> <mo>≡</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, introduced by I.&#xa0;A.&#xa0;Ibragimov. Such distributions preserve the unimodality property when it is convoluted with any unimodal distribution function. We prove that a sufficient condition of strong unimodality of a distribution is the existence of a finite first statistical moment. In particular, the class of such distributions contains absolutely continuous and logarithmically convex distributions.</p>

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STRONGLY UNIMODAL DISTRIBUTIONS WITH THE SUPPORT IN \(\mathbb {R}_+\)

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • Yu. P. Virchenko,
  • A. M. Tevolde

摘要

In this paper, we consider strongly unimodal distribution functions V whose supports lie in \(\mathbb {R}_+ \equiv [0, \infty )\) R + ≡ [ 0 , ∞ ) , introduced by I. A. Ibragimov. Such distributions preserve the unimodality property when it is convoluted with any unimodal distribution function. We prove that a sufficient condition of strong unimodality of a distribution is the existence of a finite first statistical moment. In particular, the class of such distributions contains absolutely continuous and logarithmically convex distributions.