The Annals of Statistics

Strong Approximations of the Quantile Process

Miklos Csorgo and Pal Revesz

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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.

Article information

Ann. Statist., Volume 6, Number 4 (1978), 882-894.

First available in Project Euclid: 12 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 60F15: Strong theorems

Quantile process strong approximations strong invariance Gaussian processes convergence rates


Csorgo, Miklos; Revesz, Pal. Strong Approximations of the Quantile Process. Ann. Statist. 6 (1978), no. 4, 882--894. doi:10.1214/aos/1176344261.

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