Central limit theorems for local empirical processes near boundaries of sets

Details Central limit theorems for local empirical processes near boundaries of sets Central limit theorems for local empirical processes near boundaries of sets

2011-04-21

arxiv.org/abs/1104.4220v1

Bernoulli 2011, Vol. 17, No. 2, 545-561

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/10-BEJ283 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ283

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.

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