Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables
(2015) In Journal of Theoretical Probability 28(1). p.125 Abstract
 Fix any n≥1. Let X~1,…,X~n be independent random variables. For each 1≤j≤n, X~j is transformed in a canonical manner into a random variable Xj. The Xj inherit independence from the X~j. Let sy and s∗y denote the upper 1y th −−− quantile of Sn=∑nj=1Xj and S∗n=sup1≤k≤nSk, respectively. We construct a computable quantity Q−−y based on the marginal distributions of X1,…,Xn to produce upper and lower bounds for sy and s∗y. We prove that for y≥8
6−1γ3y/16Q−−3y/16≤s∗y≤Q−−y
where
γy=12wy+1
and wy is the unique solution of
(wyeln(yy−2))wy=2y−4
for wy>ln(yy−2), and for y≥37
19γu(y)Q−−u(y)<sy≤Q−−y
where
u(y)=3y32(1+1−643y−−−−−−√).
The distribution of Sn is... (More)  Fix any n≥1. Let X~1,…,X~n be independent random variables. For each 1≤j≤n, X~j is transformed in a canonical manner into a random variable Xj. The Xj inherit independence from the X~j. Let sy and s∗y denote the upper 1y th −−− quantile of Sn=∑nj=1Xj and S∗n=sup1≤k≤nSk, respectively. We construct a computable quantity Q−−y based on the marginal distributions of X1,…,Xn to produce upper and lower bounds for sy and s∗y. We prove that for y≥8
6−1γ3y/16Q−−3y/16≤s∗y≤Q−−y
where
γy=12wy+1
and wy is the unique solution of
(wyeln(yy−2))wy=2y−4
for wy>ln(yy−2), and for y≥37
19γu(y)Q−−u(y)<sy≤Q−−y
where
u(y)=3y32(1+1−643y−−−−−−√).
The distribution of Sn is approximately centered around zero in that P(Sn≥0)≥118 and P(Sn≤0)≥165. The results extend to n=∞ if and only if for some (hence all) a>0
∑j=1∞E{(X~j−mj)2∧a2}<∞. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/1971437
 author
 Klass, Michael J and Nowicki, Krzysztof ^{LU}
 organization
 publishing date
 2015
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 quantile approximation, tail probabilities, Sum of independent random variables, tail distributions, HofmannJ/orgensen/Klass Nowicki Inequality
 in
 Journal of Theoretical Probability
 volume
 28
 issue
 1
 pages
 1  25
 publisher
 Springer
 external identifiers

 scopus:84929848604
 wos:000387922500008
 ISSN
 15729230
 DOI
 10.1007/s109590150615y
 language
 English
 LU publication?
 yes
 id
 dea1ba6490124897ba6932542c776a45 (old id 1971437)
 date added to LUP
 20160401 13:29:02
 date last changed
 20220127 19:25:27
@article{dea1ba6490124897ba6932542c776a45, abstract = {{Fix any n≥1. Let X~1,…,X~n be independent random variables. For each 1≤j≤n, X~j is transformed in a canonical manner into a random variable Xj. The Xj inherit independence from the X~j. Let sy and s∗y denote the upper 1y th −−− quantile of Sn=∑nj=1Xj and S∗n=sup1≤k≤nSk, respectively. We construct a computable quantity Q−−y based on the marginal distributions of X1,…,Xn to produce upper and lower bounds for sy and s∗y. We prove that for y≥8<br/><br> 6−1γ3y/16Q−−3y/16≤s∗y≤Q−−y<br/><br> where<br/><br> γy=12wy+1<br/><br> and wy is the unique solution of<br/><br> (wyeln(yy−2))wy=2y−4<br/><br> for wy>ln(yy−2), and for y≥37<br/><br> 19γu(y)Q−−u(y)<sy≤Q−−y<br/><br> where<br/><br> u(y)=3y32(1+1−643y−−−−−−√).<br/><br> The distribution of Sn is approximately centered around zero in that P(Sn≥0)≥118 and P(Sn≤0)≥165. The results extend to n=∞ if and only if for some (hence all) a>0<br/><br> ∑j=1∞E{(X~j−mj)2∧a2}<∞.}}, author = {{Klass, Michael J and Nowicki, Krzysztof}}, issn = {{15729230}}, keywords = {{quantile approximation; tail probabilities; Sum of independent random variables; tail distributions; HofmannJ/orgensen/Klass Nowicki Inequality}}, language = {{eng}}, number = {{1}}, pages = {{125}}, publisher = {{Springer}}, series = {{Journal of Theoretical Probability}}, title = {{Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables}}, url = {{http://dx.doi.org/10.1007/s109590150615y}}, doi = {{10.1007/s109590150615y}}, volume = {{28}}, year = {{2015}}, }