Abstract.
Let f n denote a kernel density estimator of a bounded continuous density f in the real line. Let Ψ(t) be a positive continuous function such that Under natural smoothness conditions, necessary and sufficient conditions for the sequence \scriptstyle{t\inR/}\big|\Psi(t)(fn(t)-Efn(t))\big| (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).
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1. 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, 60F15.
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Giné, E., Koltchinskii, V. & Sakhanenko, L. Kernel density estimators: convergence in distribution for weighted sup-norms. Probab. Theory Relat. Fields 130, 167–198 (2004). https://doi.org/10.1007/s00440-004-0339-x
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DOI: https://doi.org/10.1007/s00440-004-0339-x