Functional data analysis has attracted considerable interest and is facing new challenges, one of which is the increasingly available data in a streaming manner. In this article we develop an online nonparametric method to dynamically update the estimates of mean and covariance functions for functional data. The kernel-type estimates can be decomposed into two sufficient statistics depending on the data-driven bandwidths. We propose to approximate the future optimal bandwidths by a sequence of dynamically changing candidates and combine the corresponding statistics across blocks to form the updated estimation. The proposed online method is easy to compute based on the stored sufficient statistics and the current data block. We derive the asymptotic normality and, more importantly, the relative efficiency lower bounds of the online estimates of mean and covariance functions. This provides insight into the relationship between estimation accuracy and computational cost driven by the length of candidate bandwidth sequence. Simulations and real data examples are provided to support such findings.

翻译：功能数据分析引起了相当大的兴趣,并正面临新的挑战,其中之一是以流方式提供的数据越来越多。在本篇文章中,我们开发了一种在线非参数方法,以动态更新功能数据平均值和共变函数的估计数。内核型估计数可以根据数据驱动的带宽而分解成两个足够的统计数据。我们提议通过动态变化的候选人序列来估计未来的最佳带宽,并将各个区段的相应统计数据结合起来,以形成更新的估计。拟议的在线方法很容易根据储存的充足统计数据和当前数据块进行计算。我们从中和共变函数的在线估计数中得出无症状的正常性,更重要的是,相对效率较低。这样可以深入了解估计准确性与由候选带宽序列长度驱动的计算成本之间的关系。提供了模拟和真实数据实例,以支持这些调查结果。