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.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)
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, Hofmann-J/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
- 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
- 2016-04-01 13:29:02
- date last changed
- 2022-01-27 19:25:27
@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>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 = {{1572-9230}},
keywords = {{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 = {{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/s10959-015-0615-y}},
doi = {{10.1007/s10959-015-0615-y}},
volume = {{28}},
year = {{2015}},
}