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A characterization of the normal distribution by the independence of a pair of random vectors and a property of the noncentral chi-square statistic

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  • Cook, Lyle

Abstract

It is known that if the statistic Y = [Sigma]j=1n(Xj + aj)2 is drawn from a population which is distributed N(0, [sigma]) then the distribution of Y depends on only. Kagan and Shalaevski [2] have shown that if the random variables X1, X2, ..., Xn are independent and identically distributed and the distribution of Y depends only on , then each Xj is distributed N(0, [sigma]). It is shown below that if the random vectors (X1, ..., Xm) and (Xm+1, ..., Xn) are independent and the distribution of Y depends only on , then all Xj are independent and distributed N(0, [sigma]).

Suggested Citation

  • Cook, Lyle, 1971. "A characterization of the normal distribution by the independence of a pair of random vectors and a property of the noncentral chi-square statistic," Journal of Multivariate Analysis, Elsevier, vol. 1(4), pages 457-460, December.
    • Handle: RePEc:eee:jmvana:v:1:y:1971:i:4:p:457-460
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    Cited by:

    1. Ejsmont, Wiktor, 2016. "A characterization of the normal distribution by the independence of a pair of random vectors," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 1-5.

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