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Beta‐Bernstein Smoothing for Regression Curves with Compact Support
Author(s) -
Brown Bruce M.,
Chen Song Xi
Publication year - 1999
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/1467-9469.00136
Subject(s) - mathematics , estimator , smoothing , boundary (topology) , kernel (algebra) , kernel smoother , generalization , kernel density estimation , statistics , kernel method , mathematical analysis , support vector machine , combinatorics , radial basis function kernel , artificial intelligence , computer science
. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non‐parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n −4/5 , equivalent to that of standard kernel estimators when the curve has an unbounded support.