Abstract

We consider estimation of mean and covariance functions of functional snippets, which are short segments of functions possibly observed irregularly on an individual specific subinterval that is much shorter than the entire study interval. Estimation of the covariance function for functional snippets is challenging since information for the far off-diagonal regions of the covariance structure is completely missing. We address this difficulty by decomposing the covariance function into a variance function component and a correlation function component. The variance function can be effectively estimated nonparametrically, while the correlation part is modeled parametrically, possibly with an increasing number of parameters, to handle the missing information in the far off-diagonal regions. Both theoretical analysis and numerical simulations suggest that this hybrid strategy is effective. In addition, we propose a new estimator for the variance of measurement errors and analyze its asymptotic properties. This estimator is required for the estimation of the variance function from noisy measurements. Supplementary materials for this article are available online.

摘要

我们考虑估计功能片段的均值和协方差函数，这些函数片段是可能在比整个研究区间短得多的单个特定子区间上不规则地观察到的函数的短片段。功能片段的协方差函数的估计具有挑战性，因为协方差结构的远非对角线区域的信息完全缺失。我们通过将协方差函数分解为方差函数分量和相关函数分量来解决这个困难。方差函数可以非参数地有效估计，而相关部分可以参数化地建模，可能具有越来越多的参数，以处理远非对角线区域中的缺失信息。理论分析和数值模拟都表明这种混合策略是有效的。此外，我们提出了一种新的测量误差方差估计量并分析了其渐近特性。从噪声测量中估计方差函数需要该估计器。本文的补充材料可在线获取。