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Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case
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Michael J Klass, University of California Departments of Mathematics and Statistics Berkeley, CA
Krzysztof Nowicki, Lund University Department of Statistics Box 743 S-220 07 Lund, Sweden
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Abstract
Let X1, X2,... be independent and symmetric random variables
such that Sn=X1+...+Xn converges to a finite valued random variable S a.s.
and let (which is finite a.s.).
We construct upper and lower bounds for sy and sy*,
the upper 1/y-th quantile of Sy and S*, respectively.
Our approximations rely on an explicitly computable quantity
qy
for which we prove that| 0.5 qy/2<sy*<2 q2y and 0.5 q0.25 y(1+√1-8/y) <sy <2 q2y. |
The RHS's hold for y ≥2 and the LHS's for y ≥ 94 and y ≥97, respectively. Although our results are derived primarily for symmetric
random variables, they apply to non-negative variates and extend to an absolute
value of a sum of independent but otherwise arbitrary random variables.
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Full text: PDF
Pages: 1276-1298
Published on: October 16, 2007
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Electronic Journal of Probability. ISSN: 1083-6489 |
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