Change point tests for abrupt changes in the mean of functional data, i.e., random elements in infinite-dimensional Hilbert spaces, are either based on dimension reduction techniques, e.g., based on principal components, or directly based on a functional CUSUM (cumulative sum) statistic. The former have often been criticized as not being fully functional and losing too much information. On the other hand, unlike the latter, they take the covariance structure of the data into account by weighting the CUSUM statistics obtained after dimension reduction with the inverse covariance matrix. In this paper, as a middle ground between these two approaches, we propose an alternative statistic that includes the covariance structure with an offset parameter to produce a scale-invariant test procedure and to increase power when the change is not aligned with the first components. We obtain the asymptotic distribution under the null hypothesis for this new test statistic, allowing for time dependence of the data. Furthermore, we introduce versions of all three test statistics for gradual change situations, which have not been previously considered for functional data, and derive their limit distribution. Further results shed light on the asymptotic power behavior for all test statistics under various ground truths for the alternatives.

翻译：针对函数数据（即无限维希尔伯特空间中的随机元素）均值突变点的检验方法，主要基于两类技术：一类采用降维方法（如基于主成分分析），另一类直接基于函数型CUSUM（累积和）统计量。前者常被批评为非完全功能性方法，且会丢失过多信息。然而，与后者不同，这类方法通过对降维后的CUSUM统计量进行逆协方差矩阵加权，从而考虑了数据的协方差结构。本文作为这两种方法的折中方案，提出了一种替代统计量：该统计量通过引入偏移参数来纳入协方差结构，从而构建尺度不变的检验流程，并在变化未与前几个主成分对齐时提升检验功效。我们推导了该新检验统计量在零假设下的渐近分布，并允许数据存在时间依赖性。此外，我们针对函数数据中尚未被研究的渐变变化情形，提出了全部三种检验统计量的改进版本，并推导了其极限分布。进一步的研究结果揭示了所有检验统计量在不同备择假设设定下的渐近功效特性。