Gradient-based smoothing parameter selection for nonparametric regression estimation

Optimal smoothing in nonparametric conditional quantile derivative function estimation

“…The optimal h 0,cubic based on minimizing CV LCB,0 (h) is et al (2015) In this section, we show that the results of the bandwidth ratio h 0,opt /h 0,cubic derived in Henderson et al (2015) for conditional mean regression can be further simplified, and it coincides with the results in our paper for conditional quantile regression. The two terms V 1 and V 1,3 , which characterize the bandwidth ratio h 0,opt /h 0,cubic , are given in Eqs.…”

“…In this paper, we borrow the idea from Henderson et al (2015) to devise an alternative procedure for the selection of bandwidth, denoted byĥ 0,opt , which is fully data-driven without selection of initial bandwidth, and it is equivalent to the infeasible h 0,opt in the sense that it nearly minimizes the CV 0 (h) in (3), i.e.,ĥ 0,opt /h 0,opt = 1 + o P (1).…”

“…The two ratio values for the Gaussian and Epanechnikov kernels are the same as those in Henderson et al (2015) which considers nonparametric estimation of derivatives of conditional mean function. This coincidence occurs not only for these two specific kernel functions but also for any other kernel functions, as we show in the Appendix that the expression for the ratio (V 1 /V 1,3 ) 1/7 given in (16) is the same as that in Henderson et al (2015).…”

“…Borrowing the idea from Henderson et al (2015), we propose to use local cubic derivative estimator to replace the unknown true gradient in the oracle LSCV setup for the gradient in Müller et al (1987) with the local linear estimator. After this modification, the leading bias term in cross validation (CV) function is unchanged as the bias from local cubic estimator has a much smaller order than local linear estimator; and the leading variance in CV function becomes the variance of difference between the two estimators.…”

Nonparametric estimation in economics: Bayesian and frequentist approaches

“…It turns out that gradient estimates obtained from , using a bandwidth determined through least‐squares cross‐validation is (asymptotically) too small for estimating , and a rate adjustment is necessary. As an alternative, Henderson et al develop a cross‐validation function where minimization is based on the gradient of the unknown function. This approach will provide optimal smoothing if interest hinges on higher order terms in the Taylor approximation necessary for the local‐polynomial estimator.…”