The aim of this paper is to extend worst risk minimization, also called worst average loss minimization, to the functional realm. This means finding a functional regression representation that will be robust to future distribution shifts on the basis of data from two environments. In the classical non-functional realm, structural equations are based on a transfer matrix $B$. In section~\ref{sec:sfr}, we generalize this to consider a linear operator $\mathcal{T}$ on square integrable processes that plays the the part of $B$. By requiring that $(I-\mathcal{T})^{-1}$ is bounded -- as opposed to $\mathcal{T}$ -- this will allow for a large class of unbounded operators to be considered. Section~\ref{sec:worstrisk} considers two separate cases that both lead to the same worst-risk decomposition. Remarkably, this decomposition has the same structure as in the non-functional case. We consider any operator $\mathcal{T}$ that makes $(I-\mathcal{T})^{-1}$ bounded and define the future shift set in terms of the covariance functions of the shifts. In section~\ref{sec:minimizer}, we prove a necessary and sufficient condition for existence of a minimizer to this worst risk in the space of square integrable kernels. Previously, such minimizers were expressed in terms of the unknown eigenfunctions of the target and covariate integral operators (see for instance \cite{HeMullerWang} and \cite{YaoAOS}). This means that in order to estimate the minimizer, one must first estimate these unknown eigenfunctions. In contrast, the solution provided here will be expressed in any arbitrary ON-basis. This completely removes any necessity of estimating eigenfunctions. This pays dividends in section~\ref{sec:estimation}, where we provide a family of estimators, that are consistent with a large sample bound. Proofs of all the results are provided in the appendix.

翻译：本文旨在将最坏风险最小化（亦称最坏平均损失最小化）推广至函数型领域。这意味着基于来自两个环境的数据，寻找一种对未来分布偏移具有鲁棒性的函数型回归表示。在经典的非函数型领域中，结构方程基于一个转移矩阵 $B$。在章节~\ref{sec:sfr}中，我们将其推广为考虑一个作用于平方可积过程上的线性算子 $\mathcal{T}$，该算子扮演 $B$ 的角色。通过要求 $(I-\mathcal{T})^{-1}$ 有界——而非 $\mathcal{T}$ 本身有界——这将允许考虑一大类无界算子。章节~\ref{sec:worstrisk} 考察了两种不同的情况，它们均导向相同的最坏风险分解。值得注意的是，该分解具有与非函数型情形相同的结构。我们考虑任何使 $(I-\mathcal{T})^{-1}$ 有界的算子 $\mathcal{T}$，并依据偏移的协方差函数来定义未来偏移集。在章节~\ref{sec:sec:minimizer} 中，我们证明了在平方可积核空间中，该最坏风险极小化元存在的一个充要条件。此前，此类极小化元需通过目标变量与协变量积分算子的未知特征函数来表达（例如参见 \cite{HeMullerWang} 和 \cite{YaoAOS}）。这意味着为了估计极小化元，必须首先估计这些未知的特征函数。相比之下，本文提供的解将在任意正交规范基中表达。这完全消除了估计特征函数的必要性。这在章节~\ref{sec:estimation} 中带来了显著优势，我们在此提出一族估计量，它们具有大样本界的一致性。所有结果的证明均在附录中给出。