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Uniform Bounds on the Relative Error in the Approximation of Upper Quantiles for Sums of Arbitrary Independent Random Variables

Klass, Michael J and Nowicki, Krzysztof LU (2015) In Journal of Theoretical Probability 28(1). p.1-25
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)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
quantile approximation, tail probabilities, Sum of independent random variables, tail distributions, Hofmann-J/orgensen/Klass- Nowicki Inequality
in
Journal of Theoretical Probability
volume
28
issue
1
pages
1 - 25
publisher
Kluwer
external identifiers
  • scopus:84929848604
ISSN
1572-9230
DOI
10.1007/s10959-015-0615-y
language
English
LU publication?
yes
id
dea1ba64-9012-4897-ba69-32542c776a45 (old id 1971437)
date added to LUP
2011-06-09 12:00:19
date last changed
2017-01-01 05:41:53
@article{dea1ba64-9012-4897-ba69-32542c776a45,
  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&gt;ln(yy−2), and for y≥37<br/><br>
19γu(y)Q−−u(y)&lt;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&gt;0<br/><br>
∑j=1∞E{(X~j−mj)2∧a2}&lt;∞.},
  author       = {Klass, Michael J and Nowicki, Krzysztof},
  issn         = {1572-9230},
  keyword      = {quantile approximation,tail probabilities,Sum of independent random variables,tail distributions,Hofmann-J/orgensen/Klass- Nowicki Inequality},
  language     = {eng},
  number       = {1},
  pages        = {1--25},
  publisher    = {Kluwer},
  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/s10959-015-0615-y},
  volume       = {28},
  year         = {2015},
}