To reduce the dimensionality of the functional covariate, functional principal component analysis plays a key role, however, there is uncertainty on the number of principal components. Model averaging addresses this uncertainty by taking a weighted average of the prediction obtained from a set of candidate models. In this paper, we develop an optimal model averaging approach that selects the weights by minimizing a $K$-fold cross-validation criterion. We prove the asymptotic optimality of the selected weights in terms of minimizing the excess final prediction error, which greatly improves the usual asymptotic optimality in terms of minimizing the final prediction error in the literature. When the true regression relationship belongs to the set of candidate models, we provide the consistency of the averaged estimators. Numerical studies indicate that in most cases the proposed method performs better than other model selection and averaging methods, especially for extreme quantiles.

翻译：为降低函数型协变量的维数，函数主成分分析发挥着关键作用，但主成分个数存在不确定性。模型平均化通过对一组候选模型预测结果进行加权平均来应对这种不确定性。本文提出一种最优模型平均方法，通过最小化$K$折交叉验证准则选择权重。我们证明了所选权重在最小化超额最终预测误差方面的渐近最优性，这显著改进了文献中通常以最小化最终预测误差为目标的渐近最优性。当真实回归关系属于候选模型集时，我们给出了平均估计量的一致性。数值研究表明，在大多数情况下，所提方法优于其他模型选择与平均方法，尤其在极端分位数情形下表现更佳。