In this paper, we study the estimation of the derivative of a regression function in a standard univariate regression model. The estimators are defined either by derivating nonparametric least-squares estimators of the regression function or by estimating the projection of the derivative. We prove two simple risk bounds allowing to compare our estimators. More elaborate bounds under a stability assumption are then provided. Bases and spaces on which we can illustrate our assumptions and first results are both of compact or non compact type, and we discuss the rates reached by our estimators. They turn out to be optimal in the compact case. Lastly, we propose a model selection procedure and prove the associated risk bound. To consider bases with a non compact support makes the problem difficult.

翻译：本文研究标准单变量回归模型中回归函数导数的估计问题。所考虑的估计量有两种定义方式：一是通过对回归函数的非参数最小二乘估计量求导得到，二是通过直接估计导数的投影得到。我们证明了两个简单的风险界，用以比较不同估计量。随后，在稳定性假设下，我们进一步推导了更精细的风险界。可用于验证假设与初始结果的基函数和空间既包含紧支集类型，也包含非紧支集类型，并讨论了我们估计量所达到的收敛速度。在紧支集情形下，这些收敛速度被证明是最优的。最后，我们提出了一种模型选择程序，并证明了其相关的风险界。考虑具有非紧支集的基函数使得问题变得困难。