This paper develops a new automatic and location-adaptive procedure for estimating regression in a Functional Single-Index Model (FSIM). This procedure is based on $k$-Nearest Neighbours ($k$NN) ideas. The asymptotic study includes results for automatically data-driven selected number of neighbours, making the procedure directly usable in practice. The local feature of the $k$NN approach insures higher predictive power compared with usual kernel estimates, as illustrated in some finite sample analysis. As by-product we state as preliminary tools some new uniform asymptotic results for kernel estimates in the FSIM model.

翻译：本文提出了一种新的自动且位置自适应的函数型单指标模型（FSIM）回归估计方法。该过程基于$k$-最近邻（$k$NN）思想。渐近研究包括对自动数据驱动选择的邻居数量的结果，使该方法可直接应用于实践。与通常的核估计相比，$k$NN方法的局部特征确保了更高的预测能力，这一点在有限样本分析中得到了说明。作为副产品，我们初步建立了一些FSIM模型中核估计的新的均匀渐近结果。