Statistical Papers

, Volume 53, Issue 1, pp 1–21 | Cite as

Consistency of the kernel density estimator: a survey

Survey Article


Various consistency proofs for the kernel density estimator have been developed over the last few decades. Important milestones are the pointwise consistency and almost sure uniform convergence with a fixed bandwidth on the one hand and the rate of convergence with a fixed or even a variable bandwidth on the other hand. While considering global properties of the empirical distribution functions is sufficient for strong consistency, proofs of exact convergence rates use deeper information about the underlying empirical processes. A unifying character, however, is that earlier and more recent proofs use bounds on the probability that a sum of random variables deviates from its mean.


Kernel estimation Pointwise consistency Strong uniform consistency Empirical process Rate of convergence Variable bandwidth 

Mathematics Subject Classification (2000)

60-02 62-02 


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Institut für Wirtschafts- und SozialstatistikTechnische Universität DortmundDortmundGermany
  2. 2.Lehrstuhl für Statistik, Institut für VolkswirtschaftslehreUniversität RostockRostockGermany

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