This paper is concerned with adaptive nonparametric estimation using the Goldenshluger-Lepski selection method. This estimator selection method is based on pairwise comparisons between estimators with respect to some loss function. The method also involves a penalty term that typically needs to be large enough in order that the method works (in the sense that one can prove some oracle type inequality for the selected estimator). In the case of density estimation with kernel estimators and a quadratic loss, we show that the procedure fails if the penalty term is chosen smaller than some critical value for the penalty: the minimal penalty. More precisely we show that the quadratic risk of the selected estimator explodes when the penalty is below this critical value while it stays under control when the penalty is above this critical value. This kind of phase transition phenomenon for penalty calibration has already been observed and proved for penalized model selection methods in various contexts but appears here for the first time for the Goldenshluger-Lepski pairwise comparison method. Some simulations illustrate the theoretical results and lead to some hints on how to use the theory to calibrate the method in practice.
本文讨论函数协变量的估计问题。更准确地说，我们的目标是估计回归函数$m$的连续结果$Y$对标准维纳协过程$W$。继Cadre and Truquet（2015）和Cadre，Klutchnikoff，and Massiot（2017）之后，$m(W)$的Wiener-Itô分解被用于构造估计量族。在特定的光滑类的极小极大收敛速率。根据Goldenshluger和Lepski（2011）的想法定义了数据驱动的选择程序。得到了一个预言型不等式，并由此得到了自适应结果。
