TY - JOUR
T1 - Cross-validation for choosing resolution level for nonlinear wavelet curve estimators
AU - Hall, Peter
AU - Penev, Spiridon
PY - 2001
Y1 - 2001
N2 - We show that unless the target density is particularly smooth, cross-validation applied directly to nonlinear wavelet estimators produces an empirical value of primary resolution which fails, by an order of magnitude, to give asymptotic optimality. We note, too, that in the same setting, but for different reasons, cross-validation of the linear component of a wavelet estimator fails to give asymptotic optimality, if the primary resolution level that it suggests is applied to the nonlinear form of the estimator. We propose an alternative technique, based on multiple cross-validation of the linear component. Our method involves dividing the region of interest into a number of subregions, choosing a resolution level by cross-validation of the linear part of the estimator in each subregion, and taking the final empirically chosen level to be the minimum of the subregion values. This approach exploits the relative resistance of wavelet methods to over-smoothing: the final resolution level is too small in some parts of the main region, but that has a relatively minor effect on performance of the final estimator. The fact that we use the same resolution level throughout the region, rather than a different level in each subregion, means that we do not need to splice together different estimates and remove artificial jumps where the subregions abut.
AB - We show that unless the target density is particularly smooth, cross-validation applied directly to nonlinear wavelet estimators produces an empirical value of primary resolution which fails, by an order of magnitude, to give asymptotic optimality. We note, too, that in the same setting, but for different reasons, cross-validation of the linear component of a wavelet estimator fails to give asymptotic optimality, if the primary resolution level that it suggests is applied to the nonlinear form of the estimator. We propose an alternative technique, based on multiple cross-validation of the linear component. Our method involves dividing the region of interest into a number of subregions, choosing a resolution level by cross-validation of the linear part of the estimator in each subregion, and taking the final empirically chosen level to be the minimum of the subregion values. This approach exploits the relative resistance of wavelet methods to over-smoothing: the final resolution level is too small in some parts of the main region, but that has a relatively minor effect on performance of the final estimator. The fact that we use the same resolution level throughout the region, rather than a different level in each subregion, means that we do not need to splice together different estimates and remove artificial jumps where the subregions abut.
KW - Curve estimation
KW - Density estimation
KW - Generalized kernel methods
KW - Kernel estimator
KW - Leastsquares cross-validation
KW - Linear wavelet estimator
KW - Nonparametric regression
KW - Primary resolution level
KW - Thresholding
UR - http://www.scopus.com/inward/record.url?scp=33749829427&partnerID=8YFLogxK
U2 - 10.2307/3318741
DO - 10.2307/3318741
M3 - Article
SN - 1350-7265
VL - 7
SP - 317
EP - 341
JO - Bernoulli
JF - Bernoulli
IS - 2
ER -