The KLR-Theorem Revisited

Abram Kagan ()
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Abram Kagan: University of Maryland

Sankhya A: The Indian Journal of Statistics, 2021, vol. 83, issue 2, No 2, 549-553

Abstract: Abstract For independent random variables X1,…,Xn;Y1,…,Yn with all Xi identically distributed and same for Yj, we study the relation E { a X ̄ + b Y ̄ | X 1 − X ̄ + Y 1 − Y ̄ , … , X n − X ̄ + Y n − Y ̄ } = const $$ E\{a\bar X + b\bar Y|X_{1} -\bar X +Y_{1} -\bar Y,\ldots,X_{n} -\bar X +Y_{n} -\bar Y\}=\text{const} $$ with a,b some constants. It is proved that for n ≥ 3 and ab > 0 the relation holds iff Xi and Yj are Gaussian. A new characterization arises in case of a = 1,b = − 1. In this case either Xi or Yj or both have a Gaussian component. It is the first (at least known to the author) case when presence of a Gaussian component is a characteristic property.

Keywords: Constancy of regression-Characterization; Gaussian component; Primary 62H05; Secondary 62J02 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13171-019-00183-2

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