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Probab. Theory Relat. Fields 130, 167198 (2004)Evarist Gine Vladimir Koltchinskii Lyudmila Sakhanenko
Received: 2 July 2002 / Revised version: 24 December 2003 /
Published online: 3 March 2004 c[circlecopyrt]Digital Object Identifier (DOI) 10.1007/s00440-004-0339-xKernel density estimators: convergence
in distribution for weighted sup-normsSpringer-Verlag 2004Abstract. Let fn denote a kernel density estimator of a bounded continuous density f in
the real line. Let (t) be a positive continuous function such that f <. Under natural smoothness conditions, necessary and sufficient conditions for the sequence [radicalbigg] nhn2log h1nsuptR [vextendsingle][vextendsingle](t)(fn(t)Efn(t))[vextendsingle][vextendsingle](properly centered and normalized) to converge in distribution to the
double exponential law are obtained. The proof is based on Gaussian approximation and a
(new) limit theorem for weighted sup-norms of a stationary Gaussian process. This extends
well known results of Bickel and Rosenblatt to the case of weighted sup-norms, with the
sup taken over the whole line. In addition, all other possible limit distributions of the above
sequence are identified (subject to some regularity assumptions).1. IntroductionWe consider the kernel density estimator fn of an unknown density f in the real
line based on a sample (X1,...,Xn) of size n of i.i.d. observations with density
f , kernel K and bandwidth hn such that hn 0 and nhn as n :fn(t) =1nhn
n
i=1K
. (1.1)Our main goal is to study the convergence in distribution of the (properly centered and normalized) sequenceE. Gine: Departments of Mathematics and Statistics, University of Connecticut, Storrs, CT
06269-3009, USA. e-mail: [email protected]. Koltchinskii: Department of Mathematics and Statistics, University of New Mexico Albuquerque, NM 87131-1141, USA. e-mail: [email protected]. Sakhanenko: Department of Statistics and Probability, Michigan State University East
Lansing, MI 48824-1027, USA. e-mail: [email protected]. Research partially supported by NSF Grant No. DMS-0070382.2. Research partially supported by NSA Grant No. MDA904-02-1-0075 and NSF Grant No.
DMS-0304861.Mathematics Subject Classification (2000): Primary 62G07; secondary 62G20, 60F15Key words and phrases: Kernel density estimator Convergence in distribution Weighted
sup-normt Xi
hn168 E. Gine et al.[radicalBigg] nhn2 log h1n[vextenddouble]
[vextenddouble]
[vextenddouble]()(fn Efn)()[vextenddouble](1.2)
where is a positive weight function that might depend on the density f (for
instance, it might be f with some > 0). In particular, under some regularity
assumptions, we prove the following main result. Letr(t) := [integraltext]
K(u)K(u + t)duK2.Suppose it satisfies the conditionr(t) =1 C|t|+ o(|t|) (1.3)for some...