Existing functional data analysis literature has mostly overlooked data with spikes in mean, such as weekly sporting goods sales by a salesperson which spikes around holidays. For such functional data, two-step estimation procedures are formulated for the population mean function and holiday effect parameters, which correspond to the population sales curve and the spikes in sales during holiday times. The estimators are based on spline smoothing for individual trajectories using non-holiday observations, and are shown to be oracally efficient in the sense that both the mean function and holiday effects are estimated as efficiently as if all individual trajectories were known a priori. Consequently, an asymptotic simultaneous confidence band is established for the mean function and confidence intervals for holiday effects, respectively. Two sample extensions are also formulated and simulation experiments provide strong evidence that corroborates the asymptotic theory. Application to sporting goods sales data has led to a number of new discoveries.

中文翻译：

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具有假日效应的密集功能数据的口头有效估计

现有的功能数据分析文献大多忽略了均值出现峰值的数据，例如，营业员每周在假日前后出现峰值的体育用品销售。对于此类功能数据，针对人口均值函数和假日效应参数制定了两步估算程序，这些参数与人口销售曲线和假日期间的销售峰值相对应。估计器基于使用非假日观测值对单个轨迹进行样条平滑处理，并且在平均功能和假日效应都被高效估计的意义上被证明是口头上有效的，就好像所有单个轨迹都是先验已知的一样。因此，分别为假日效应的平均函数和置信区间建立了渐近同时置信带。还制定了两个样本扩展，并且仿真实验提供了强有力的证据，证实了渐近理论。在体育用品销售数据中的应用带来了许多新发现。