The Annals of Statistics

Convergence of Reduced Empirical and Quantile Processes with Application to Functions of Order Statistics in the Non-I.I.D. Case

Galen R. Shorack

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Abstract

Any triangular array of row independent $\mathrm{rv}$'s having continuous $\mathrm{df}$'s can be transformed naturally so that the empirical and quantile processes of the resulting $\mathrm{rv}$'s are relatively compact. Moreover, convergence (to a necessarily normal process) takes place if and only if a simple covariance function converges pointwise. Using these results we derive the asymptotic normality of linear combinations of functions of order statistics of non-i.i.d. $\mathrm{rv}$'s in the case of bounded scores.

Article information

Source
Ann. Statist., Volume 1, Number 1 (1973), 146-152.

Dates
First available in Project Euclid: 25 October 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1193342391

Digital Object Identifier
doi:10.1214/aos/1193342391

Mathematical Reviews number (MathSciNet)
MR336776

Zentralblatt MATH identifier
0255.62044

Subjects
Primary: 62G30: Order statistics; empirical distribution functions
Secondary: 60B10: Convergence of probability measures 62E20: Asymptotic distribution theory

Keywords
Convergence of non-i.i.d. empirical processes asymptotic normality of functions of non-i.i.d. order statistics

Citation

Shorack, Galen R. Convergence of Reduced Empirical and Quantile Processes with Application to Functions of Order Statistics in the Non-I.I.D. Case. Ann. Statist. 1 (1973), no. 1, 146--152. doi:10.1214/aos/1193342391. https://projecteuclid.org/euclid.aos/1193342391


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