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Oracally efficient estimation for dense functional data with holiday effects. (English) Zbl 1439.62189
Summary: 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.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G08 Nonparametric regression and quantile regression 62P20 Applications of statistics to economics 62R10 Functional data analysis 65D07 Numerical computation using splines 62G15 Nonparametric tolerance and confidence regions
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