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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.

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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[1] Anzanello, M.; Fogliatto, F., Learning curve models and applications: literature review and research directions, Int J Ind Ergon, 41, 573-583 (2011)
[2] Benko, M.; Härdle, W.; Kneip, A., Common functional principal components, Ann Statist, 37, 1-34 (2009) · Zbl 1169.62057
[3] Bosq, D., Linear processes in function spaces: theory and applications (2000), New York: Springer, New York · Zbl 0962.60004
[4] Cai, L.; Yang, L., A smooth simultaneous confidence band for conditional variance function, TEST, 24, 632-655 (2015) · Zbl 1327.62194
[5] Cai, L.; Liu, R.; Wang, S.; Yang, L., Simultaneous confidence bands for mean and variance functions based on deterministic design, Stat Sin, 29, 505-525 (2019) · Zbl 1412.62213
[6] Cao, G.; Wang, L.; Li, Y.; Yang, L., Oracle efficient confidence envelopes for covariance functions in dense functional data, Stat Sin, 26, 359-383 (2016) · Zbl 1419.62071
[7] Cao, G.; Yang, L.; Todem, D., Simultaneous inference for the mean function based on dense functional data, J Nonparametr Statist, 24, 359-377 (2012) · Zbl 1241.62119
[8] Cardot, H., Nonparametric estimation of smoothed principal components analysis of sampled noisy functions, J Nonparametr Stat, 12, 503-538 (2000) · Zbl 0951.62030
[9] Cho, H.; Fryzlewicz, P., Multiple-change-point detection for high dimensional time series via sparsified binary segmentation, J R Stat Soc B, 77, 475-507 (2015) · Zbl 1414.62356
[10] Claeskens, G.; Van Keilegom, I., Bootstrap confidence bands for regression curves and their derivatives, Ann Stat, 31, 1852-1884 (2003) · Zbl 1042.62044
[11] De Boor, C., A practical guide to splines (1978), New York: Springer, New York · Zbl 0406.41003
[12] Degras, D., Simultaneous confidence bands for nonparametric regression with functional data, Stat Sin, 21, 1735-1765 (2011) · Zbl 1225.62052
[13] Fan, J.; Huang, T.; Li, R., Analysis of longitudinal data with semiparametric estimation of covariance function, J Am Stat Assoc, 102, 632-642 (2007) · Zbl 1172.62323
[14] Fan, J.; Lin, S., Tests of significance when data are curves, J Am Stat Assoc, 93, 1007-1021 (1998) · Zbl 1064.62525
[15] Fan, J.; Zhang, W., Simultaneous confidence bands and hypothesis testing in varying coefficient models, Scand J Stat, 27, 715-731 (2000) · Zbl 0962.62032
[16] Ferraty, F.; Vieu, P., Nonparametric functional data analysis: theory and practice (2006), New York: Springer, New York · Zbl 1119.62046
[17] Fryzlewicz, P.; Subba Rao, S., Multiple-change-point detection for auto-regressive conditional heteroscedastic processes, J R Stat Soc B, 76, 903-924 (2014) · Zbl 1411.62248
[18] Gu, L.; Wang, L.; Härdle, W.; Yang, L., A simultaneous confidence corridor for varying coefficient regression with sparse functional data, TEST, 23, 806-843 (2014) · Zbl 1312.62051
[19] Gu, L.; Yang, L., Oracally efficient estimation for single-index link function with simultaneous confidence band, Electron J Stat, 9, 1540-1561 (2015) · Zbl 1327.62254
[20] Hall, P.; Müller, H.; Wang, J., Properties of principal component methods for functional and longitudinal data analysis, Ann Stat, 34, 1493-1517 (2006) · Zbl 1113.62073
[21] Huang, J.; Yang, L., Identification of nonlinear additive autoregressive models, J R Stat Soc B, 66, 463-477 (2004) · Zbl 1062.62185
[22] Huang, X.; Wang, L.; Yang, L.; Kravchenko, A., Management practice effects on relationships of grain yields with topography and precipitation, Agron J, 100, 1463-1471 (2008)
[23] James, G.; Hastie, T.; Sugar, C., Principal component models for sparse functional data, Biometrika, 87, 587-602 (2000) · Zbl 0962.62056
[24] James, G.; Sugar, C., Clustering for sparsely sampled functional data, J Am Stat Assoc, 98, 397-408 (2003) · Zbl 1041.62052
[25] Komlós, J.; Major, P.; Tusnády, G., An approximation of partial sums of independent RV’s, and the sample DF II, Z. Wahrscheinlichkeitstheorie Verw. Gebiete, 34, 33-58 (1976) · Zbl 0307.60045
[26] Li, B.; Yu, Q., Classification of functional data: a segmentation approach, Comput Stat Data Anal, 52, 4790-4800 (2008) · Zbl 1452.62992
[27] Ma, S.; Yang, L.; Carroll, Rj, A simultaneous confidence band for sparse longitudinal regression, Stat Sin, 22, 95-122 (2012) · Zbl 1417.62088
[28] Ma, S., A plug-in the number of knots selector for polynomial spline regression, J Nonparametr Stat, 26, 489-507 (2014) · Zbl 1305.62151
[29] Raña, P.; Aneiros, G.; Vilar, Jm, Detection of outliers in functional time series, Environmetrics, 26, 178-191 (2015)
[30] Rice, J.; Wu, C., Nonparametric mixed effects models for unequally sampled noisy curves, Biometrics, 57, 253-259 (2001) · Zbl 1209.62061
[31] Schröder, Al; Fryzlewicz, P., Adaptive trend estimation in financial time series via multiscale change-point-induced basis recovery, Stat Interface, 6, 449-461 (2013) · Zbl 1326.91035
[32] Song, Q.; Yang, L., Spline confidence bands for variance function, J Nonparametric Stat, 21, 589-609 (2009) · Zbl 1165.62317
[33] Wang, J.; Liu, R.; Cheng, F.; Yang, L., Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band, Ann Stat, 42, 654-668 (2014) · Zbl 1308.62096
[34] Wang, J.; Wang, S.; Yang, L., Simultaneous confidence bands for the distribution function of a finite population and of its superpopulation, TEST, 25, 692-709 (2016) · Zbl 1391.62077
[35] Wang, J.; Yang, L., Polynomial spline confidence bands for regression curves, Stat Sin, 19, 325-342 (2009) · Zbl 1225.62055
[36] Wu, W.; Zhao, Z., Inference of trends in time series, J R Stat Soc B, 69, 391-410 (2007)
[37] Yao, F.; Müller, H.; Wang, J., Functional data analysis for sparse longitudinal data, J Am Stat Assoc, 100, 577-590 (2005) · Zbl 1117.62451
[38] Zhang, J., Analysis of variance for functional data (2013), Boca Raton: Chapman & Hall/CRC, Boca Raton
[39] Zhao, Z.; Wu, W., Confidence bands in nonparametric time series regression, Ann Stat, 36, 1854-1878 (2008) · Zbl 1142.62346
[40] Zheng, S.; Liu, R.; Yang, L.; Hädle, W., Statistical inference for generalized additive models: simultaneous confidence corridors and variable selection, TEST, 25, 607-626 (2016) · Zbl 1422.62155
[41] Zheng, S.; Yang, L.; Härdle, W., A smooth simultaneous confidence corridor for the mean of sparse functional data, J Am Stat Assoc, 109, 661-673 (2014) · Zbl 1367.62151
[42] Zhou, S.; Shen, X.; Wolfe, D., Local asymptotics of regression splines and confidence regions, Ann Stat, 26, 1760-1782 (1998) · Zbl 0929.62052
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