Olga Y. Savchuk, Jeffrey D. Hart and Simon P. Sheather. (2013) “One-Sided Cross-Validation for Nonsmooth Regression Functions”, Journal of Nonparametric Statistics, 25:4.
When the above article was first published online, there was an error in the 
expansion. In particular, it is wrong to claim that in the case of the kernel K satisfying conditions (C1)–(C6),
In fact,
Thus, the expansion (9) must be corrected so that the constant defined by (10) is replaced by the constant defined as
The difference between and is only in the limits of integration with respect to z. Indeed, in the integration is performed over the interval , whereas in the case of the region of integration is . Notice that the interval of integration in covers the whole support of K, whereas in the case of the support of K is, essentially, cut.
In fact, the expansion (9) and the constant (10) are correct in the case when K has support , which is the case for the Epanechnikov and quartic kernels. However, for K satisfying (C1)–C(6), the constant must be used. The proof of this fact is available from the authors.
The corrected expansion matches that of of the kernel density estimator in the article of Cline and Hart (1991) after the density function is replaced by the regression function and the term is eliminated. This further validates our correction.
The corresponding corrections must also be made in the expansion based on the one-sided kernel L satisfying conditions (L1)–(L6). Indeed, the appropriate constant in the expansion is
After the corrections, the expressions for and become invariant to rescaling of K and L, respectively. Indeed, consider
The above depends on K through . The constant is invariant to rescaling of K in that for , . Similarly, is invariant to rescaling of L.
The nonsmooth rescaling constant defined by (13) updates to the following:
The last row of Table 1, corresponding to the Gaussian kernel, must be replaced by the following:
Thus,
R is less than 1% for all kernels in Table 1, including the Gaussian kernel.
In Section 5, the kernel with and is not robust after adjusting our computation of the nonsmooth rescaling constant. Indeed, has the smooth rescaling constant , whereas its corrected nonsmooth rescaling constant is . It appears that . Thus, has even greater discrepancy between the rescaling constants compared to the Gaussian kernel. This again raises the question of whether or not there are exactly robust kernels.
As noted in the article, the value of is very close to 0.5284, that is the (wrong) nonsmooth rescaling constant in the case when the Gaussian kernel is used in both cross-validation and estimation stages of the method. Since the one-sided kernel based on selects practically the same bandwidths as the one-sided Gaussian kernel, the boxplots in Figure 2 for in the cases of the nonsmooth functions and can be identified with those produced by the one-sided Gaussian kernel along with the rescaling constant of 0.5284. Then, the negative bandwidth bias evident from the boxplots is explained by an inappropriately low rescaling constant. Using 0.5730 instead of 0.5284 would provide the appropriate centering of the corresponding boxplots. Thus, empirical evidence supports the corresponding theoretical findings.
A couple of other corrections:
The expression for on p. 892 lacks an additive term . Thus, the corrected expression is
The condition on p. 902 that as is not sufficient for the outlined proof. A minimum sufficient condition is that .
Assumption (D4) should be updated so that f has two continuous derivatives on the interval .
The authors apologize for the errors.
