Electronic Journal of Statistics

Uniform convergence and asymptotic confidence bands for model-assisted estimators of the mean of sampled functional data

Hervé Cardot, Camelia Goga, and Pauline Lardin

Full-text: Open access

Abstract

When the study variable is functional and storage capacities are limited or transmission costs are high, selecting with survey sampling techniques a small fraction of the observations is an interesting alternative to signal compression techniques, particularly when the goal is the estimation of simple quantities such as means or totals. We extend, in this functional framework, model-assisted estimators with linear regression models that can take account of auxiliary variables whose totals over the population are known. We first show, under weak hypotheses on the sampling design and the regularity of the trajectories, that the estimator of the mean function as well as its variance estimator are uniformly consistent. Then, under additional assumptions, we prove a functional central limit theorem and we assess rigorously a fast technique based on simulations of Gaussian processes which is employed to build asymptotic confidence bands. The accuracy of the variance function estimator is evaluated on a real dataset of sampled electricity consumption curves measured every half an hour over a period of one week.

Article information

Source
Electron. J. Statist. Volume 7 (2013), 562-596.

Dates
First available in Project Euclid: 14 March 2013

Permanent link to this document
http://projecteuclid.org/euclid.ejs/1363268498

Digital Object Identifier
doi:10.1214/13-EJS779

Mathematical Reviews number (MathSciNet)
MR3035266

Zentralblatt MATH identifier
06167948

Subjects
Primary: 62L20: Stochastic approximation
Secondary: 60F05: Central limit and other weak theorems

Keywords
Calibration covariance function functional linear model GREG Hájek estimator Horvitz-Thompson estimator linear interpolation survey sampling

Citation

Cardot, Hervé; Goga, Camelia; Lardin, Pauline. Uniform convergence and asymptotic confidence bands for model-assisted estimators of the mean of sampled functional data. Electron. J. Statist. 7 (2013), 562--596. doi:10.1214/13-EJS779. http://projecteuclid.org/euclid.ejs/1363268498.


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References

  • Adler, R. J. and Taylor, J. E. (2007). Random Fields and Geometry. Springer-Verlag, New York.
  • Bhatia, R. (1997). Matrix Analysis. Springer-Verlag, New York.
  • Billingsley, P. (1968). Convergence of Probability Measures. John Wiley and Sons.
  • Boistard, H., Lopuhaä, H. P., and Ruiz-Gazen, A. (2012). Approximation of rejective sampling inclusion probabilites and application to higher order correlation. Electronic J. of Statistics, 6:1967–1983.
  • Bosq, D. (2000). Linear Processes in Function Spaces: Theory and Applications, volume 149 of Lecture notes in Statistics. Springer-Verlag, New York.
  • Breidt, F. J. and Opsomer, J. D. (2000). Local polynomial regression estimators in survey sampling. Ann. Statist., 28(4):1023–1053.
  • Callado, A., Kamienski, C., Szabó, G., Gerö, B., Kelner, J., Fernandes, S. and Sadok, D. (2009). A Survey on Internet Traffic Identification and Classification. IEEE Communications Surveys and Tutorials, 11:37–52.
  • Cardot, H. (2007). Conditional functional principal components analysis. Scandinavian J. of Statistics, 34:317–335.
  • Cardot, H., Chaouch, M., Goga, C., and Labruère, C. (2010). Properties of design-based functional principal components analysis. J. of Statistical Planning and Inference, 140:75–91.
  • Cardot, H., Degras, D., and Josserand, E. (2012a). Confidence bands for Horvitz-Thompson estimators using sampled noisy functional data. To appear in Bernoulli.
  • Cardot, H., Dessertaine, A., Goga, C., Josserand, E., and Lardin, P. (2012b). Comparaison de différents plans de sondage et construction de bandes de confiance pour l’estimation de la moyenne de données fonctionnelles: une illustration sur la consommation électrique. To appear in Techniques d’Enquêtes/Survey Methodology.
  • Cardot, H., Goga, C., and Lardin, P. (2012c). Variance estimation and asymptotic confidence bands for the mean estimator of sampled functional data with high entropy unequal probability sampling designs. Arxiv:1209.6503
  • Cardot, H. and Josserand, E. (2011). Horvitz-Thompson estimators for functional data: asymptotic confidence bands and optimal allocation for stratified sampling. Biometrika, 98:107–118.
  • Chaouch, M., and Goga, C. (2012). Using complex surveys to estimate the L1-median of a functional variable: application to electricity load curves. International Statistical Review, 80:40–59.
  • Chiou, J., Müller, H., and Wang, J. (2004). Functional response models. Statistica Sinica, 14:675–693.
  • Cuevas, A., Febrero, M., and Fraiman, R. (2006). On the use of the bootstrap for estimating functions with functional data. Computational Statistics and Data Analysis, 51:1063–1074.
  • Degras, D. (2011). Simultaneous confidence bands for parametric regression with functional data. Statistica Sinica, 21(4):1735–1765.
  • Degras, D. (2012). Rotation sampling for functional data. http://arxiv.org/abs/1204.4494.
  • Deville, J. C. and Särndal, C. E. (1992). Calibration estimators in survey sampling. J. Amer. Statist. Assoc., 87:376–382.
  • Faraway, J. (1997). Regression analysis for a functional response. Technometrics, 39(3):254–261.
  • Ferraty, F., Laksaci, A., Tadj, A., and Vieu, P. (2011). Kernel regression with functional response. Electronic J. of Statist., 5:159–171.
  • Fuller, W. A. (2009). Sampling Statistics. John Wiley and Sons, Hoboken, New Jersey.
  • Goga, C., and Ruiz-Gazen, A. (2013). Efficient estimation of nonlinear finite population parameters using nonparametrics. Journal of the Royal Statistical Society, Series B, to appear.
  • Guillas, S. (2001). Rates of convergence of autocorrelation estimates for autoregressive Hilbertian processes. Statist. and Probability Letters, 55:281–291.
  • Hahn, M. (1977). Conditions for sample-continuity and the central limit theorem. Annals of Probability, 5:351–360.
  • Hájek, J. (1981). Sampling From a Finite Population. Statistics: Textbooks and Monographs. Marcel Dekker, New York.
  • Isaki, C. and Fuller, W. (1982). Survey design under the regression superpopulation model. J. Amer. Statist. Assoc., 77:49–61.
  • Pitt, L. D. and Tran, L. T. (1979). Local sample path properties of Gaussian fields. Annals of Probability, 7:477–493.
  • Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis. Springer-Verlag, New York, second edition.
  • Robinson, P. and Särndal, C. (1983). Asymptotic properties of the generalized regression estimator in probability sampling. Sankhya: The Indian Journal of Statistics, 45:240–248.
  • Särndal, C. (1980). On $\pi$ inverse weighting versus best linear unbiased weighting in probability sampling. Biometrika, 67:639–50.
  • Särndal, C. E., Swensson, B., and Wretman, J. (1992). Model Assisted Survey Sampling. Springer series in statistics. Springer-Verlag, New York.
  • van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.
  • van der Vaart, A. W. and Wellner, J. A. (2000). Weak Convergence and Empirical Processes. With Applications to Statistics. Springer-Verlag, New York.