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Data depth for measurable noisy random functions. (English) Zbl 1415.62028

Summary: In the literature on data depth applicable to random functions, it is usually assumed that the trajectories of all the random curves are continuous, known at each point of the domain, and observed exactly. These assumptions turn out to be unrealistic in practice, as the functions are often observed only on a finite grid of time points, and in the presence of measurement errors. In this work, we provide the necessary theoretical background enabling the extension of the statistical methodology based on data depth to measurable (not necessarily continuous) random functions observed within the latter framework. It is shown that even if the random functions are discontinuous, observed discretely, and contaminated with additive noise, many common depth functionals maintain the fine consistency properties valid in the ideal case of completely observed noiseless functions. For the integrated depth for functions, we provide uniform rates of convergence over the space of integrable functions.

MSC:

62G20 Asymptotic properties of nonparametric inference
62G35 Nonparametric robustness
62G08 Nonparametric regression and quantile regression
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[1] Agostinelli, C.; Romanazzi, M., Local depth, J. Statist. Plann. Inference, 141, 817-830 (2011) · Zbl 1353.62019
[2] Apostol, T. M., Introduction to Analytic Number Theory (1976), Springer-Verlag: Springer-Verlag New York · Zbl 0335.10001
[3] Billingsley, P., Convergence of Probability Measures (1999), Wiley: Wiley New York · Zbl 0172.21201
[4] Bogachev, V. I., Measure Theory, Vol. II (2007), Springer-Verlag: Springer-Verlag Berlin · Zbl 1120.28001
[5] Bott, A.-K.; Devroye, L.; Kohler, M., Estimation of a distribution from data with small measurement errors, Electron. J. Stat., 7, 2457-2476 (2013) · Zbl 1293.62068
[6] J.L.O. Cabrera, R; J.L.O. Cabrera, R
[7] Chakraborty, A.; Chaudhuri, P., The spatial distribution in infinite dimensional spaces and related quantiles and depths, Ann. Statist., 42, 1203-1231 (2014) · Zbl 1305.62141
[8] Claeskens, G.; Hubert, M.; Slaets, L.; Vakili, K., Multivariate functional halfspace depth, J. Amer. Statist. Assoc., 109, 411-423 (2014) · Zbl 1367.62162
[9] Crambes, C.; Kneip, A.; Sarda, P., Smoothing splines estimators for functional linear regression, Ann. Statist., 37, 35-72 (2009) · Zbl 1169.62027
[10] Cuevas, A., A partial overview of the theory of statistics with functional data, J. Statist. Plann. Inference, 147, 1-23 (2014) · Zbl 1278.62012
[11] Cuevas, A.; Febrero, M.; Fraiman, R., On the use of the bootstrap for estimating functions with functional data, Comput. Statist. Data Anal., 51, 1063-1074 (2006) · Zbl 1157.62390
[12] Cuevas, A.; Febrero, M.; Fraiman, R., Robust estimation and classification for functional data via projection-based depth notions, Comput. Statist., 22, 481-496 (2007) · Zbl 1195.62032
[13] Cuevas, A.; Fraiman, R., On depth measures and dual statistics: A methodology for dealing with general data, J. Multivariate Anal., 100, 753-766 (2009) · Zbl 1163.62039
[14] De Brabanter, K.; Cao, F.; Gijbels, I.; Opsomer, J., Local polynomial regression with correlated errors in random design and unknown correlation structure, Biometrika, 105, 681-690 (2018) · Zbl 1499.62128
[15] Dudley, R. M., Uniform Central Limit Theorems (1999), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0951.60033
[16] Dudley, R. M., Real Analysis and Probability (2002), Cambridge University Press: Cambridge University Press Cambridge, Revised reprint of the 1989 original · Zbl 1023.60001
[17] Dutta, S.; Ghosh, A. K.; Chaudhuri, P., Some intriguing properties of Tukey’s half-space depth, Bernoulli, 17, 1420-1434 (2011) · Zbl 1229.62063
[18] Eichelsbacher, S.; Schmock, U., Rank-dependent moderate deviations of U-empirical measures in strong topologies, Probab. Theory Related Fields, 126, 61-90 (2003) · Zbl 1039.60023
[19] Fan, J.; Gijbels, I., Local Polynomial Modelling and its Applications (1996), Chapman & Hall: Chapman & Hall London · Zbl 0873.62037
[20] Ferraty, F.; Vieu, P., Nonparametric Functional Data Analysis: Theory and Practice (2006), Springer: Springer New York · Zbl 1119.62046
[21] Fraiman, R.; Muniz, G., Trimmed means for functional data, Test, 10, 419-440 (2001) · Zbl 1016.62026
[22] Genton, M. G.; Johnson, C.; Potter, K.; Stenchikov, G.; Sun, Y., Surface boxplots, Stat, 3, 1-11 (2014)
[23] Gijbels, I.; Nagy, S., Consistency of non-integrated depths for functional data, J. Multivariate Anal., 140, 259-282 (2015) · Zbl 1327.62305
[24] Goia, A.; Vieu, P., An introduction to recent advances in high/infinite dimensional statistics, J. Multivariate Anal., 146, 1-6 (2016) · Zbl 1384.00073
[25] Grinblat, L.Š., A limit theorem for measurable random processes and its applications, Proc. Amer. Math. Soc., 61, 371-376 (1976) · Zbl 0379.60009
[26] Györfi, L.; Kohler, M.; Krzyzak, A.; Walk, H., A Distribution-free Theory of Nonparametric Regression (2002), Springer-Verlag: Springer-Verlag New York · Zbl 1021.62024
[27] Horváth, L.; Kokoszka, P., Inference for Functional Data With Applications (2012), Springer: Springer New York · Zbl 1279.62017
[28] Hsing, T.; Eubank, R., Theoretical Foundations of Functional Data Analysis, With an Introduction to Linear Operators (2015), Wiley: Wiley Chichester · Zbl 1338.62009
[29] Hyndman, R. J.; Shang, H. L., Rainbow plots, bagplots, and boxplots for functional data, J. Comput. Graph. Statist., 19, 29-45 (2010)
[30] Ieva, F.; Paganoni, A. M., Depth measures for multivariate functional data, Comm. Statist. Theory Methods, 42, 1265-1276 (2013) · Zbl 1347.62093
[31] Ivanov, A. V., Convergence of distributions of functionals of measurable random fields, Ukrain. Mat. Zh., 32, 27-34 (1980), 141
[32] Jiang, C.-R.; Wang, J.-L., Covariate adjusted functional principal components analysis for longitudinal data, Ann. Statist., 38, 1194-1226 (2010) · Zbl 1183.62102
[33] Kohler, M.; Krzyzak, A.; Walk, H., Optimal global rates of convergence for nonparametric regression with unbounded data, J. Statist. Plann. Inference, 139, 1286-1296 (2009) · Zbl 1153.62031
[34] Li, Y.; Hsing, T., Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data, Ann. Statist., 38, 3321-3351 (2010) · Zbl 1204.62067
[35] Liu, R. Y., On a notion of data depth based on random simplices, Ann. Statist., 18, 405-414 (1990) · Zbl 0701.62063
[36] Liu, X.-H., Kernel Smoothing for Spatially Correlated Data (2001), Department of Statistics, Iowa State University: Department of Statistics, Iowa State University Ames, IA, (Ph.D. thesis)
[37] López-Pintado, S.; Jornsten, R., Functional analysis via extensions of the band depth, (Complex Datasets and Inverse Problems (2007), Inst. Math. Statist.: Inst. Math. Statist. Beachwood, OH), 103-120
[38] López-Pintado, S.; Romo, J., On the concept of depth for functional data, J. Amer. Statist. Assoc., 104, 718-734 (2009) · Zbl 1388.62139
[39] López-Pintado, S.; Romo, J., A half-region depth for functional data, Comput. Statist. Data Anal., 55, 1679-1695 (2011) · Zbl 1328.62029
[40] López-Pintado, S.; Sun, Y.; Lin, J.; Genton, M. G., Simplicial band depth for multivariate functional data, Adv. Data Anal. Classif., 8, 321-338 (2014) · Zbl 1414.62066
[41] López-Pintado, S.; Wei, Y., Depth for sparse functional data, (Recent Advances in Functional Data Analysis and Related Topics. Recent Advances in Functional Data Analysis and Related Topics, Contrib. Statist. (2011), Physica-Verlag/Springer: Physica-Verlag/Springer Heidelberg), 209-212
[42] Mosler, K., Depth statistics, (Becker, C.; Fried, R.; Kuhnt, S., Robustness and Complex Data Structures (2013), Springer: Springer Heidelberg), 17-34
[43] M. Mosler, Y. Polyakova, General notions of depth for functional data, arXiv preprint arXiv:1208.1981; M. Mosler, Y. Polyakova, General notions of depth for functional data, arXiv preprint arXiv:1208.1981
[44] Müller, H.-G.; Wu, Y.; Yao, F., Continuously additive models for nonlinear functional regression, Biometrika, 100, 607-622 (2013) · Zbl 1284.62407
[45] Nagy, S., Consistency of h-mode depth, J. Statist. Plann. Inference, 165, 91-103 (2015) · Zbl 1326.62108
[46] Nagy, S., Integrated depth for measurable functions and sets, Statist. Probab. Lett., 123, 165-170 (2017) · Zbl 1360.62236
[47] Nagy, S., An overview of consistency results for depth functionals, (Aneiros, G.; Bongiorno, E. G.; Cao, R.; Vieu, P., Functional Statistics and Related Fields (2017), Springer International Publishing: Springer International Publishing Cham, Switzerland), 189-196
[48] Nagy, S.; Gijbels, I.; Hlubinka, D., Weak convergence of discretely observed functional data with applications, J. Multivariate Anal., 146, 46-62 (2016) · Zbl 1334.62090
[49] Nagy, S.; Gijbels, I.; Omelka, M.; Hlubinka, D., Integrated depth for functional data: Statistical properties and consistency, ESAIM Probab. Stat., 20, 95-130 (2016) · Zbl 1357.62201
[50] Narisetty, N. N.; Nair, V. N., Extremal depth for functional data and applications, J. Amer. Statist. Assoc., 111, 1705-1714 (2016)
[51] Opsomer, J.; Wang, Y.; Yang, Y., Nonparametric regression with correlated errors, Statist. Sci., 16, 134-153 (2001) · Zbl 1059.62537
[52] Paul, D.; Peng, J., Consistency of restricted maximum likelihood estimators of principal components, Ann. Statist., 37, 1229-1271 (2009) · Zbl 1161.62032
[53] O. Pokotylo, P. Mozharovskyi, R. Dyckerhoff, S. Nagy, R; O. Pokotylo, P. Mozharovskyi, R. Dyckerhoff, S. Nagy, R
[54] RRR; RRR
[55] Radchenko, P.; Qiao, X.; James, G. M., Index models for sparsely sampled functional data, J. Amer. Statist. Assoc., 110, 824-836 (2015) · Zbl 1373.62346
[56] Ramsay, J. O.; Silverman, B. W., Functional Data Analysis (2015), Springer: Springer New York · Zbl 1079.62006
[57] Stone, C. J., Consistent nonparametric regression, Ann. Statist., 5, 595-645 (1977) · Zbl 0366.62051
[58] Stone, C. J., Optimal global rates of convergence for nonparametric regression, Ann. Statist., 10, 1040-1053 (1982) · Zbl 0511.62048
[59] Sun, Y.; Genton, M. G., Functional boxplots, J. Comput. Graph. Statist., 20, 316-334 (2011)
[60] Sun, Y.; Genton, M. G., Adjusted functional boxplots for spatio-temporal data visualization and outlier detection, Environmetrics, 23, 54-64 (2012)
[61] Tukey, J. W., Mathematics and the picturing of data, (Proceedings of the International Congress of Mathematicians (Vancouver, BC, 1974), Vol. 2 (1975), Canad. Math. Congress: Canad. Math. Congress Montréal, QC), 523-531
[62] Varadarajan, V. S., On the convergence of sample probability distributions, Sankhyā, 19, 23-26 (1958) · Zbl 0082.34201
[63] Whitaker, R. T.; Mirzargar, M.; Kirby, R. M., Contour boxplots: A method for characterizing uncertainty in feature sets from simulation ensembles, IEEE Trans. Vis. Comput. Graphics, 19, 2713-2722 (2013)
[64] Wu, Y.; Fan, J.; Müller, H.-G., Varying-coefficient functional linear regression, Bernoulli, 16, 730-758 (2010) · Zbl 1220.62046
[65] Yao, F.; Müller, H.-G.; Wang, J.-L., Functional data analysis for sparse longitudinal data, J. Amer. Statist. Assoc., 100, 577-590 (2005) · Zbl 1117.62451
[66] Zhang, X.; Wang, J.-L., From sparse to dense functional data and beyond, Ann. Statist., 44, 2281-2321 (2016) · Zbl 1349.62161
[67] Zuo, Y.; Serfling, R. J., General notions of statistical depth function, Ann. Statist., 28, 461-482 (2000) · Zbl 1106.62334
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