On difference-based gradient estimation in nonparametric regression

We propose a framework to directly estimate the gradient in multivariate nonparametric regression models that bypasses fitting the regression function. Specifically, we construct the estimator as a linear combination of adjacent observations with the coefficients from a vector-valued difference sequence, so it is more flexible than existing methods. Under the equidistant designs, closed-form solutions of the optimal sequences are derived by minimizing the estimation variance, with the estimation bias well controlled. We derive the theoretical properties of the estimators and show that they achieve the optimal convergence rate. Further, we propose a data-driven tuning parameter-selection criterion for practical implementation. The effectiveness of our estimators is validated via simulation studies and a real data application.