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Asymptotic normality of lightly trimmed means – a converse

Published online by Cambridge University Press:  24 October 2008

R. A. Maller
R. A. Maller
Affiliation:
CSIRO, Division of Mathematics and Statistics, Wembley, Western Australia
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Extract

Let Xi be iidrv's and Sn = X1 + X2 + … + Xn. Let (r)Sn be Sn minus the r terms of largest absolute value. If (Sn/Bn)− An converges to normality for some Bn, An, then so does ((r)Sn/Bn) − An. We show that the converse is true if Xi have a continuous symmetric distribution, and give some related results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 3 , November 1982 , pp. 535 - 545
Copyright
Copyright © Cambridge Philosophical Society 1982

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References

REFERENCES

(1)Arov, D. Z. and Bobrov, A. A.The extreme terms of a sample and their role in the sum of independent variables. Theor. Prob. Appl. 5 (1960), 377–396.CrossRefGoogle Scholar
(2)Darling, D.The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 (1952), 95–107.CrossRefGoogle Scholar
(3)Feller, W.An introduction to probability theory and its applications, vol. 2, 2nd ed. (Wiley, N.Y., 1966).Google Scholar
(4)Gnedenko, B. V. and Kolmogorov, A. N.Limit distributions for sums of independent ran-dom variables, 2nd ed. (Addison and Wesley, Reading, Mass., 1968).Google Scholar
(5)Hall, P.On the extreme terms of a sample from the domain of attraction of a stable law. J. Lond. Math. Soc. 18 (1978), 181–191.CrossRefGoogle Scholar
(6)Hatori, H., Maejima, M. and Mori, T.Convergence rates in the law of large numbers when extreme terms are excluded. Z. Wahrscheinlichkeitstheorie verw. Geb. 47 (1979), 1–12.CrossRefGoogle Scholar
(7)Loève, M.Probability theory, 3rd ed. (Van Nostrand, Princeton, N. J., 1963).Google Scholar
(8)Maller, R. A.Some properties of stochastic compactness. J. Aust. Math. Soc. A 30 (1981), 264–277.CrossRefGoogle Scholar
(9)Mori, T.The strong law of large numbers when extreme terms are excluded from sums. Z. Wahrscheinlichkeitstheorie verw. Geb. 36 (1976), 189–194.CrossRefGoogle Scholar
(10)Stigler, S. M.The asymptotic distribution of the trimmed mean. Ann. Statist. 1 (1973), 472–477.CrossRefGoogle Scholar