TY - JOUR
T1 - Boundary performance of the beta kernel estimators
AU - Zhang, Shunpu
AU - Karunamuni, Rohana J.
N1 - Funding Information:
The authors are grateful to the editor and the two referees for their helpful comments which have led to significant improvement of the paper. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada.
PY - 2010/1
Y1 - 2010/1
N2 - The beta kernel estimators are shown in Chen [S.X. Chen, Beta kernel estimators for density functions, Comput. Statist. Data Anal. 31 (1999), pp. 131-145] to be non-negative and have less severe boundary problems than the conventional kernel estimator. Numerical results in Chen [S.X. Chen, Beta kernel estimators for density functions, Comput. Statist. Data Anal. 31 (1999), pp. 131-145] further show that beta kernel estimators have better finite sample performance than some of the widely used boundary corrected estimators. However, our study finds that the numerical comparisons of Chen are confounded with the choice of the bandwidths and the quantities being compared. In this paper, we show that the performances of the beta kernel estimators are very similar to that of the reflection estimator, which does not have the boundary problem only for densities exhibiting a shoulder at the endpoints of the support. For densities not exhibiting a shoulder, we show that the beta kernel estimators have a serious boundary problem and their performances at the boundary are inferior to that of the well-known boundary kernel estimator.
AB - The beta kernel estimators are shown in Chen [S.X. Chen, Beta kernel estimators for density functions, Comput. Statist. Data Anal. 31 (1999), pp. 131-145] to be non-negative and have less severe boundary problems than the conventional kernel estimator. Numerical results in Chen [S.X. Chen, Beta kernel estimators for density functions, Comput. Statist. Data Anal. 31 (1999), pp. 131-145] further show that beta kernel estimators have better finite sample performance than some of the widely used boundary corrected estimators. However, our study finds that the numerical comparisons of Chen are confounded with the choice of the bandwidths and the quantities being compared. In this paper, we show that the performances of the beta kernel estimators are very similar to that of the reflection estimator, which does not have the boundary problem only for densities exhibiting a shoulder at the endpoints of the support. For densities not exhibiting a shoulder, we show that the beta kernel estimators have a serious boundary problem and their performances at the boundary are inferior to that of the well-known boundary kernel estimator.
KW - Beta kernel estimator
KW - Boundary kernel estimator
KW - Boundary problem
KW - Nonparametric density estimation
KW - Reflection estimator
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U2 - 10.1080/10485250903124984
DO - 10.1080/10485250903124984
M3 - Article
AN - SCOPUS:77049101016
SN - 1048-5252
VL - 22
SP - 81
EP - 104
JO - Journal of Nonparametric Statistics
JF - Journal of Nonparametric Statistics
IS - 1
ER -