Open Access
Translator Disclaimer
April, 1978 Linear Bounds on the Empirical Distribution Function
Galen R. Shorack, Jon A. Wellner
Ann. Probab. 6(2): 349-353 (April, 1978). DOI: 10.1214/aop/1176995582


Let $\Gamma_n$ denote the empirical df of a sample from the uniform (0, 1) df $I$. Let $\xi_{nk}$ denote the $k$th smallest observation. Let $\lambda_n > 1$. Let $A_n$ denote the event that $\Gamma_n$ intersects the line $\lambda_n I$ on [0, 1] and let $B_n$ denote the event that $\Gamma_n$ intersects the line $I/\lambda_n$ on $\lbrack\xi_{n1}, 1\rbrack$. Conditions on $\lambda_n$ are given that determine whether $P(A_n \mathrm{i.o.})$ and $P(B_n \mathrm{i.o.})$ equal 0 or 1. Results for $A_n$ (for $B_n$) are related to upper class sequences for $1/(n\xi_{n1})$ (for $n\xi_{n2})$. Upper class sequences for $n\xi_{nk}$, with $k > 1$, are characterized. In the case of nonidentically distributed random variables, we present the result analogous to $P(A_n \mathrm{i.o.}) = 0$.


Download Citation

Galen R. Shorack. Jon A. Wellner. "Linear Bounds on the Empirical Distribution Function." Ann. Probab. 6 (2) 349 - 353, April, 1978.


Published: April, 1978
First available in Project Euclid: 19 April 2007

MathSciNet: MR467893
zbMATH: 0376.60034
Digital Object Identifier: 10.1214/aop/1176995582

Primary: 60F15
Secondary: 60G17 , 62G30

Keywords: $k$th smallest order statistic , empirical process , linear bounds , non i.i.d. case , upper class characterizations

Rights: Copyright © 1978 Institute of Mathematical Statistics


Vol.6 • No. 2 • April, 1978
Back to Top