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Uniform strong consistency of sample quantiles

✍ Scribed by Ryszard Zieliński

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
215 KB
Volume
37
Category
Article
ISSN
0167-7152

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✦ Synopsis

It is well known that if xq(F) is the unique qth quantile of a distribution function F, then Xk(,):, with k(n)/n--~ q is a strongly consistent estimator of xq(F). However, for every e > 0 and for every, even very large n, suPFe: ~ PF{IXk<,):, -xq(F)l >/3} = 1. This is a consequence of the fact that in the family of all distribution functions with uniquely defined qth quantile the almost sure convergence Xk(,):,--~ xq(F) is not uniform. A simple necessary and sufficient condition for the uniform strong consistency of X, tn):, is given.

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