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<p>For a sample of size two from a continuous density <i>f</i>(<i>x</i>) with support on the positive half-line <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\((0,\infty )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the distributions of the sum of the two variables is usually of a different character from the distribution of the maximum of the two variables. It can be verified that, surprisingly, in the case in which the variables are half-normal, the distributions of the sum and of the maximum are of the same type. After discussing two alternative elementary proofs of this half-normal property, we show that, under mild regularity conditions, it only occurs if the two i.i.d. variables have a common half-normal distribution. The analogous case in which samples of size greater than 2 are involved is also discussed.</p>

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On a Characteristic Property of the Half-Normal Distribution

  • 期刊论文

  • 发表日期：

    2026年2月17日

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  • Barry C. Arnold,
  • Jose A. Villasenor

摘要

For a sample of size two from a continuous density f(x) with support on the positive half-line \((0,\infty )\) ( 0 , ∞ ) , the distributions of the sum of the two variables is usually of a different character from the distribution of the maximum of the two variables. It can be verified that, surprisingly, in the case in which the variables are half-normal, the distributions of the sum and of the maximum are of the same type. After discussing two alternative elementary proofs of this half-normal property, we show that, under mild regularity conditions, it only occurs if the two i.i.d. variables have a common half-normal distribution. The analogous case in which samples of size greater than 2 are involved is also discussed.