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On the quadratic moment of self-normalized sums
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Mathematical Statistics.
2010 (English)In: Statistics and Probability Letters, ISSN 0167-7152, Vol. 80, no 17-18, 1289-1296 p.Article in journal (Refereed) Published
Abstract [en]

Let an integer n >= 2 and a vector of independent, identically distributed random variables X-1, ..., X-n be given with P(X = 0) = 0 and define the self-normalized sum Z(n) = (Sigma(n)(i=1) X-i)/(Sigma(n)(i=1) X-i(2))(1/2). With a formula for EZ(n)(2) we prove that EZ(n)(2) >= 1 and that EZ(n)(2) = 1 if and only if the summands are symmetrically distributed. We also construct examples where Z(n) converges to the standard normal distribution as n tends to infinity while EZ(n)(2) tends to infinity (the distribution of the summands varies with n).

Place, publisher, year, edition, pages
2010. Vol. 80, no 17-18, 1289-1296 p.
Keyword [en]
Quadratic moment, Self-normalization, Symmetric distributions, Student's t-test
National Category
URN: urn:nbn:se:uu:diva-135426DOI: 10.1016/j.spl.2010.04.008ISI: 000280916100004OAI: oai:DiVA.org:uu-135426DiVA: diva2:375021
Available from: 2010-12-07 Created: 2010-12-06 Last updated: 2011-04-05Bibliographically approved

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Jonsson, Fredrik
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