Meny

Javascript verkar inte påslaget? - Vissa delar av Lunds universitets webbplats fungerar inte optimalt utan javascript, kontrollera din webbläsares inställningar.
Du är här

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

Författare:
Publiceringsår: 2015
Språk: Engelska
Sidor: 1-25
Publikation/Tidskrift/Serie: Journal of Theoretical Probability
Volym: 28
Nummer: 1
Dokumenttyp: Artikel i tidskrift
Förlag: Kluwer

Sammanfattning

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)
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}<∞.

Nyckelord

  • Probability Theory and Statistics
  • quantile approximation
  • tail probabilities
  • Sum of independent random variables
  • tail distributions
  • Hofmann-J/orgensen/Klass- Nowicki Inequality

Övriga

Published
  • ISSN: 1572-9230

Box 117, 221 00 LUND
Telefon 046-222 00 00 (växel)
Telefax 046-222 47 20
lu [at] lu.se

Fakturaadress: Box 188, 221 00 LUND
Organisationsnummer: 202100-3211
Om webbplatsen