Functional data present as functions or curves possessing a spatial or temporal component. These components by nature have a fixed observational domain. Consequently, any asymptotic investigation requires modelling the increased correlation among observations as density increases due to this fixed domain constraint. One such appropriate stochastic process is the Ornstein-Uhlenbeck process. Utilizing this spatial autoregressive process, we demonstrate that parameter estimators for a simple linear regression model display inconsistency in an infill asymptotic domain. Such results are contrary to those expected under the customary increasing domain asymptotics. Although none of these estimator variances approach zero, they do display a pattern of diminishing return regarding decreasing estimator variance as sample size increases. This may prove invaluable to a practitioner as this indicates perhaps an optimal sample size to cease data collection. This in turn reduces time and data collection cost because little information is gained in sampling beyond a certain sample size.

翻译：函数型数据表现为具有空间或时间成分的函数或曲线。这些成分本质上具有固定的观测域。因此，任何渐近研究都需要模拟因固定域约束而随密度增加而增强的观测间相关性。一种恰当的这种随机过程是奥恩斯坦-乌伦贝克过程。利用这一空间自回归过程，我们证明简单线性回归模型的参数估计量在填充渐近域下表现出不一致性。这一结果与常规递增域渐近下的预期结论相反。尽管这些估计量的方差均不趋近于零，但它们呈现出随着样本量增加而估计量方差递减收益递减的模式。这对实践者可能极具价值，因为这表明存在一个终止数据收集的最优样本量。由于超过一定样本量后继续采样所获信息甚微，这进而可减少时间与数据收集成本。