The Law of the Iterated Logarithm for a Triangular Array of Empirical Processes
Miguel A. Arcones, State University of New York
Abstract
We study the
compact law of the iterated logarithm for a certain type of
triangular arrays of
empirical processes, appearing in statistics
(M-estimators, regression, density estimation, etc).
We give necessary and sufficient conditions
for the law of the iterated logarithm of these
processes of the type of conditions used in Ledoux and Talagrand (1991):
convergence in probability, tail conditions and total boundedness of the
parameter space with respect to certain pseudometric.
As an application, we consider the law of
the iterated logarithm for a class of density estimators. We obtain
the order of the optimal window for the law of the iterated
logarithm of density estimators.
We also consider the compact law of the iterated logarithm
for kernel density estimators when they have large
deviations similar to those of a
Poisson process.
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