The Annals of Probability

Linear Bounds on the Empirical Distribution Function

Galen R. Shorack and Jon A. Wellner

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Abstract

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 $P(A_n \mathrm{i.o.}) = 0$.

Article information

Source
Ann. Probab., Volume 6, Number 2 (1978), 349-353.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176995582

Digital Object Identifier
doi:10.1214/aop/1176995582

Mathematical Reviews number (MathSciNet)
MR467893

Zentralblatt MATH identifier
0376.60034

JSTOR
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G17: Sample path properties 62G30: Order statistics; empirical distribution functions

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

Citation

Shorack, Galen R.; Wellner, Jon A. Linear Bounds on the Empirical Distribution Function. Ann. Probab. 6 (1978), no. 2, 349--353. doi:10.1214/aop/1176995582. https://projecteuclid.org/euclid.aop/1176995582


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