Insensitivity of Nadaraya–Watson estimators to design correlation

Universal local linear kernel estimators in nonparametric regression

“…Moreover, the above-mentioned arguments and the representations ( 40) and ( 42) imply (36). Further, the first estimator in (37) is obvious by the above remark about the domain of summation in the definition of functions w nj (t), and the relations sup…”

“…The second estimator in (37) immediately follows from the well-known estimate of the error of approximation by Riemann integral sums of the corresponding integrals of smooth functions on a finite closed interval:…”

“…4). Similar conditions for design elements were also used in [37] and [40] in nonparametric regression, and in [36]- [38] -in nonlinear regression.…”

Universal Local Linear Kernel Estimators in Nonparametric Regression

“…Univariate versions of this estimation problem were studied in Borisov et al (2021) and Linke et al (2022) where the asymptotic analysis and simulations showed that the proposed estimators perform better than the Nadaraya-Watson ones in several cases. Note that the univariate case in Borisov et al (2021) does not allow direct generalization to a multivariate case, since the weights were defined there as the spacings of the variational series generated by the design elements.…”

“…One of the univariate estimators studied here may be more accurate than the estimator in Borisov et al (2021) (see Remark 3 below). Conditions on the design elements similar to those of this paper were used Linke and Borisov (2022), and in Linke (2023). The conditions provide uniform consistency of the estimators, but guarantee only pointwise consistency of the Nadarya-Watson ones.…”