Central limit theorems for local empirical processes near boundaries of sets

J.H.J. Einmahl, E.V. Khmaladze

Research output: Contribution to journalArticleScientificpeer-review

Abstract

We define the local empirical process, based on n i.i.d. random vectors in dimension d, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with n and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.
Original languageEnglish
Pages (from-to)545-561
JournalBernoulli
Volume17
Issue number2
Publication statusPublished - 2011

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Empirical Process
Central limit theorem
Shrinking
Random Vector
Vary

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title = "Central limit theorems for local empirical processes near boundaries of sets",
abstract = "We define the local empirical process, based on n i.i.d. random vectors in dimension d, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with n and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.",
author = "J.H.J. Einmahl and E.V. Khmaladze",
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Central limit theorems for local empirical processes near boundaries of sets. / Einmahl, J.H.J.; Khmaladze, E.V.

In: Bernoulli, Vol. 17, No. 2, 2011, p. 545-561.

Research output: Contribution to journalArticleScientificpeer-review

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T1 - Central limit theorems for local empirical processes near boundaries of sets

AU - Einmahl, J.H.J.

AU - Khmaladze, E.V.

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PY - 2011

Y1 - 2011

N2 - We define the local empirical process, based on n i.i.d. random vectors in dimension d, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with n and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.

AB - We define the local empirical process, based on n i.i.d. random vectors in dimension d, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical processes, indexed by classes of sets that vary with n and satisfy certain conditions, an appropriately defined uniform central limit theorem holds. The concept of differentiation of sets in measure is very convenient for developing the results. Some examples and statistical applications are also presented.

M3 - Article

VL - 17

SP - 545

EP - 561

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

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