Abstract
Estimating and constructing a simultaneous confidence band for the mean function in the presence of outliers is an important problem in the framework of functional data analysis. In this paper, we propose a robust estimator and a robust simultaneous confidence band for the mean function of functional data using M-estimation and B-splines. The robust simultaneous confidence band is also extended to the difference of mean functions of two populations. Further, the asymptotic properties of the M-based mean function estimator, such as the asymptotic consistency and asymptotic normality, are studied. The performance of the proposed robust methods and their robustness are demonstrated with an extensive simulation study and two real data examples.
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Cao’s research is supported in part by the Simons Foundation under Grant #354917 and the National Science Foundation under Grants DMS 1736470. We thank the Associate Editor and two anonymous referees for their helpful and constructive comments, which lead to significant improvement in this paper.
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Appendix: Variance of pseudo-data
Appendix: Variance of pseudo-data
In order to evaluate the efficiency of pseudo-data method, we compare the sample variance of the pseudo-data with the real variance of the uncontaminated model defined in Sect. 3.1. To further emphasize the influence of outliers in the calculation of the variance, we also compared the results with the sample variance of the outlier contaminated dataset, using the least-squares method as the estimator for the mean function. We generate a functional dataset from the model in Sect. 3.1 for sample size \(n = 200\), with \(s_l = 5\) and \(s_u = 7\) for peak outlier case. Each simulation is repeated 500 times. The results are presented in Fig. 4. We only present the results for peak outliers, as all other cases have similar results. The results show that the variance of the pseudo-data is very close to the true sample variance computed from the clean dataset, while the non-robust estimation of the variance of the contaminated dataset is strongly affected by the outlier curves.
Comparison of the real variance (solid), sample variance of pseudo-data (dotted) and sample variance of contaminated data (dashed). Peak outlier with 5% contamination
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Lima, I.R., Cao, G. & Billor, N. M-based simultaneous inference for the mean function of functional data. Ann Inst Stat Math 71, 577–598 (2019). https://doi.org/10.1007/s10463-018-0656-y
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DOI: https://doi.org/10.1007/s10463-018-0656-y