The Annals of Statistics

No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions

Luc Devroye and Laszlo Gyorfi

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Abstract

For any sequence of empirical probability measures $\{\mu_n\}$ on the Borel sets of the real line and any $\delta > 0$, there exists a singular continuous probability measure $\mu$ such that $\inf_n \sup_A |\mu_n(A) - \mu(A)| \geq \frac{1}{2} - \delta \quad\text{almost surely}.$

Article information

Source
Ann. Statist., Volume 18, Number 3 (1990), 1496-1499.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176347765

Digital Object Identifier
doi:10.1214/aos/1176347765

Mathematical Reviews number (MathSciNet)
MR1062724

Zentralblatt MATH identifier
0707.60026

JSTOR
links.jstor.org

Subjects
Primary: 62G05: Estimation
Secondary: 60E05: Distributions: general theory

Keywords
Empirical measure total variation distance singular continuous distributions

Citation

Devroye, Luc; Gyorfi, Laszlo. No Empirical Probability Measure can Converge in the Total Variation Sense for all Distributions. Ann. Statist. 18 (1990), no. 3, 1496--1499. doi:10.1214/aos/1176347765. https://projecteuclid.org/euclid.aos/1176347765


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