The Annals of Mathematical Statistics

Some Distributions of Sample Means

George W. Brown and John W. Tukey

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It is shown that certain monomials in normally distributed quantities have stable distributions with index $2^{-k}$. This provides, for $k > 1$, simple examples where the mean of a sample has a distribution equivalent to that of a fixed, arbitrarily large multiple of a single observation. These examples include distributions symmetrical about zero, and positive distributions. Using these examples, it is shown that any distribution with a very long tail (of average order $\geq x^{-3/2}$) has the distributions of its sample means grow flatter and flatter as the sample size increases. Thus the sample mean provides less information than a single value. Stronger results are proved for still longer tails.

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Ann. Math. Statist., Volume 17, Number 1 (1946), 1-12.

First available in Project Euclid: 28 April 2007

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Brown, George W.; Tukey, John W. Some Distributions of Sample Means. Ann. Math. Statist. 17 (1946), no. 1, 1--12. doi:10.1214/aoms/1177731017.

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See also

  • Later Acknowledgment of Priority: Shashanka S. Mitra. Acknowledgment of Priority: "Distribution of Symmetric Stable Laws of Index $2^{-n}$" AND "Stable Laws of Index $2^{-n}$". Ann. Probab., Vol. 11, Iss. 2 (1983), 456.