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.

中文翻译：

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非参数回归中基于差异的梯度估计

我们提出了一个框架来直接估计多元非参数回归模型中的梯度，绕过拟合回归函数。具体来说，我们将估计器构造为相邻观测值与向量值差序列中的系数的线性组合，因此它比现有方法更灵活。在等距设计下，通过最小化估计方差导出最优序列的闭式解，并且估计偏差得到很好的控制。我们推导了估计器的理论特性，并表明它们实现了最佳收敛率。此外，我们提出了一种用于实际实施的数据驱动的调整参数选择标准。我们的估算器的有效性通过模拟研究和真实数据应用得到验证。