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July, 1978 Strong Approximations of the Quantile Process
Miklos Csorgo, Pal Revesz
Ann. Statist. 6(4): 882-894 (July, 1978). DOI: 10.1214/aos/1176344261


Let $q_n(y), 0 < y < 1,$ be a quantile process based on a sequence of i.i.d. rv with distribution function $F$ and density function $f.$ Given some regularity conditions on $F$ the distance of $q_n(y)$ and the uniform quantile process $u_n(y),$ respectively defined in terms of the order statistics $X_{k:n}$ and $U_{k:n} = F(X_{k:n}),$ is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes, a law of iterated logarithm for $q_n(y)$ and, using similar results for the uniform quantile process $u_n(y),$ it is also shown that $q_n(y)$ can be approximated by a sequence of Brownian bridges as well as by a Kiefer process.


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Miklos Csorgo. Pal Revesz. "Strong Approximations of the Quantile Process." Ann. Statist. 6 (4) 882 - 894, July, 1978.


Published: July, 1978
First available in Project Euclid: 12 April 2007

zbMATH: 0378.62050
MathSciNet: MR501290
Digital Object Identifier: 10.1214/aos/1176344261

Primary: 62G30
Secondary: 60F15

Keywords: Convergence rates , Gaussian processes , quantile process , strong approximations , Strong invariance

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 4 • July, 1978
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