Abstract
A binary classification problem is considered, where the posteriori probability is estimated by the nonparametric kernel regression estimate with naive kernel. The excess error probability of the corresponding plug-in decision classification rule according to the error probability of the Bayes decision is studied such that the excess error probability is decomposed into approximation and estimation error. A general formula is derived for the approximation error. Under a weak margin condition and various smoothness conditions, tight upper bounds are presented on the approximation error. By a Berry-Esseen type central limit theorem a general expression for the estimation error is shown.
This work was supported in part by the National Development Agency (NFÜ, Hungary) as part of the project Introduction of Cognitive Methods for UAV Collision Avoidance Using Millimeter Wave Radar, (grant no.: KMR-12-1-2012-0008).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Audibert J-Y, Tsybakov AB (2007) Fast learning rates for plug-in classifiers, Ann Stat 35:608–633
Devroye L (1981) On the almost everywhere convergence of nonparametric regression function estimates. Ann Stat 9:1310–1319
Devroye L, Györfi L, Lugosi G (1996) A probabilistic theory of pattern recognition. Springer, New York
Devroye L, Wagner TJ (1980) Distribution-free consistency results in nonparametric discrimination and regression function estimation. Ann Stat 8:231–239
Györfi L, Kohler M, Krzyżak A, Walk H (2002) A distribution-free theory of nonparametric regression. Springer, New York
Hall P, Kang K (2005) Bandwidth choice for nonparametric classification. Ann Stat 33:284–306
Kohler M, Krzyżak A (2007) On the rate of convergence of local averaging plug-in classification rules under a margin condition. IEEE Trans Inf Theory 53:1735–1742
Krzyżak A (1986) The rates of convergence of kernel regression estimates and classification rules. IEEE Trans Inf Theory IT-32:668–679
Krzyżak A, Pawlak M (1984) Distribution-free consistency of a nonparametric kernel regression estimate and classification. IEEE Trans Inf Theory IT-30:78–81
Mammen E, Tsybakov AB (1999) Smooth discrimination analysis. Ann Stat 27:1808–1829
Michel R (1981) On the constant in the non-uniform version of the Berry-Esseen theorem. Z Wahrsch Verw Gebiete 55:109–117
Petrov VV (1975) Sums of independent random variables. Springer, Berlin
Tsybakov AB (2004) Optimal aggregation of classifiers in statistical learning. Ann Stat 32:135–166
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Döring, M., Györfi, L., Walk, H. (2016). Exact Rate of Convergence of Kernel-Based Classification Rule. In: Matwin, S., Mielniczuk, J. (eds) Challenges in Computational Statistics and Data Mining. Studies in Computational Intelligence, vol 605. Springer, Cham. https://doi.org/10.1007/978-3-319-18781-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-18781-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18780-8
Online ISBN: 978-3-319-18781-5
eBook Packages: EngineeringEngineering (R0)