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Characterizations of certain multivariate distributions*

Published online by Cambridge University Press:  24 October 2008

Y. H. Wang
Y. H. Wang
Affiliation:
Department of Mathematics, Sir George Williams University, Montreal 107, Canada
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Extract

Let X1, X2, …, Xn, be n (n ≥ 2) independent observations on a one-dimensional random variable X with distribution function F. Let

be the sample mean and

be the sample variance. In 1925, Fisher (2) showed that if the distribution function F is normal then and S2 are stochastically independent. This property was used to derive the student's t-distribution which has played a very important role in statistics. In 1936, Geary(3) proved that the independence of and S2 is a sufficient condition for F to be a normal distribution under the assumption that F has moments of all order. Later, Lukacs (14) proved this result assuming only the existence of the second moment of F. The assumption of the existence of moments of F was subsequently dropped in the proofs given by Kawata and Sakamoto (7) and by Zinger (27). Thus the independence of and S2 is a characterizing property of the normal distribution.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 75 , Issue 2 , March 1974 , pp. 219 - 234
Copyright
Copyright © Cambridge Philosophical Society 1974

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References

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