Integrated depth for functional data: statistical properties and consistency

Nagy, Stanislav; Gijbels, Irène; Omelka, Marek; Hlubinka, Daniel

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 95-130.

Several depths suitable for infinite-dimensional functional data that are available in the literature are of the form of an integral of a finite-dimensional depth function. These functionals are characterized by projecting functions into low-dimensional spaces, taking finite-dimensional depths of the projected quantities, and finally integrating these projected marginal depths over a preset collection of projections. In this paper, a general class of integrated depths for functions is considered. Several depths for functional data proposed in the literature during the last decades are members of this general class. A comprehensive study of its most important theoretical properties, including measurability and consistency, is given. It is shown that many, but not all, properties of the integrated depth are shared with the finite-dimensional depth that constitutes its building block. Some pending measurability issues connected with all integrated depth functionals are resolved, a broad new notion of symmetry for functional data is proposed, and difficulties with respect to consistency results are identified. A general universal consistency result for the sample depth version, and for the generalized median, for integrated depth for functions is derived.

Classification : 62G20, 62H99, 60F15
Mots clés : Center of symmetry, functional data, generalized median, integrated depth, measurability, strong consistency, weak consistency

L.D. Brown and R. Purves, Measurable selections of extrema. Ann. Statist. 1 (1973) 902–912. | DOI | MR | Zbl

G. Claeskens, M. Hubert, L. Slaets and K. Vakili, Multivariate functional halfspace depth. J. Amer. Statist. Assoc. 109 (2014) 411–423. | DOI | MR | Zbl

J.A. Cuesta-Albertos and A. Nieto-Reyes, The random Tukey depth. Comput. Statist. Data Anal. 52 (2008) 4979–4988. | DOI | MR | Zbl

A. Cuevas and R. Fraiman, On depth measures and dual statistics. A methodology for dealing with general data. J. Multivariate Anal. 100 (2009) 753–766. | DOI | MR | Zbl

A. DasGupta, Probability for statistics and machine learning: Fundamentals and advanced topics. Springer Texts in Statistics. Springer, New York (2011). | MR | Zbl

V.H. de la Peña and E. Giné, Decoupling. From dependence to independence. Probability and its Applications. Springer-Verlag, New York (1999). | MR

D.L. Donoho and M. Gasko, Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 (1992) 1803–1827. | DOI | MR | Zbl

R.M. Dudley, Uniform central limit theorems. Vol. 63 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). | MR | Zbl

R.M. Dudley, Real analysis and probability. Vol. 74 of Cambridge Studies in Advanced Mathematics. Revised reprint of the 1989 original. Cambridge University Press, Cambridge (2002). | MR | Zbl

R.M. Dudley, E. Giné and J. Zinn, Uniform and universal Glivenko-Cantelli classes. J. Theoret. Probab. 4 (1991) 485–510. | DOI | MR | Zbl

L. Dümbgen, Limit theorems for the simplicial depth. Statist. Probab. Lett. 14 (1992) 119–128. | DOI | MR | Zbl

S. Dutta, A.K. Ghosh and P. Chaudhuri, Some intriguing properties of Tukey’s half-space depth. Bernoulli 17 (2011) 1420–1434. | DOI | MR | Zbl

R. Fraiman and J. Meloche, Multivariate $L$-estimation. With comments and a rejoinder by the authors. Test 8 (1999) 255–317. | MR | Zbl

R. Fraiman and G. Muniz, Trimmed means for functional data. Test 10 (2001) 419–440. | DOI | MR | Zbl

I. Gijbels and S. Nagy, Consistency of non-integrated depths for functional data. J. Multivariate Anal. 140 (2015) 259–282. | DOI | MR | Zbl

F.R. Hampel, A general qualitative definition of robustness. Ann. Math. Statist. 42 (1971) 1887–1896. | DOI | MR | Zbl

X. He and G. Wang, Convergence of depth contours for multivariate datasets. Ann. Statist. 25 (1997) 495–504. | MR | Zbl

L. Kong and Y. Zuo, Smooth depth contours characterize the underlying distribution. J. Multivariate Anal. 101 (2010) 2222–2226. | DOI | MR | Zbl

J. Kuelbs and J. Zinn, Concerns with functional depth. ALEA Latin Am. J. Prob. Math. Statist. 10 (2013) 831–855. | MR | Zbl

E.H. Lieb and M. Loss, Analysis. Vol. 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI. 2nd edition (2001). | MR | Zbl

R.Y. Liu, On a notion of data depth based on random simplices. Ann. Statist. 18 (1990) 405–414. | MR | Zbl

R.Y. Liu, J.M. Parelius and K. Singh, Multivariate analysis by data depth: descriptive statistics, graphics and inference. With discussion and a rejoinder by Liu and Singh. Ann. Statist. 27 (1999) 783–858. | DOI | MR | Zbl

S. López-Pintado and J. Romo, On the concept of depth for functional data. J. Amer. Statist. Assoc. 104 (2009) 718–734. | DOI | MR | Zbl

S. López-Pintado and J. Romo, A half-region depth for functional data. Comput. Statist. Data Anal. 55 (2011) 1679–1695. | DOI | MR | Zbl

S. López-Pintado, Y. Sun, J. Lin and M.G. Genton, Simplicial band depth for multivariate functional data. Adv. Data Anal. Classif. 8 (2014) 321–338. | DOI | MR | Zbl

J.-C. Massé and R. Theodorescu, Halfplane trimming for bivariate distributions. J. Multivariate Anal. 48 (1994) 188–202. | DOI | MR | Zbl

I. Mizera, Qualitative robustness and weak continuity: the extreme unction? In Nonparametrics and robustness in modern statistical inference and time series analysis: a Festschrift in honor of Professor Jana Jurečková. Vol. 7 of Inst. Math. Stat. Collect. Inst. Math. Statist., Beachwood, OH (2010) 169–181. | MR

I. Mizera and M. Volauf, Continuity of halfspace depth contours and maximum depth estimators: diagnostics of depth-related methods. J. Multivariate Anal. 83 (2002) 365–388. | DOI | MR | Zbl

K. Mosler, Multivariate dispersion, central regions and depth: The lift zonoid approach. Vol. 165 of Lect. Notes Stat. Springer-Verlag, Berlin (2002). | MR | Zbl

K. Mosler, Depth statistics. Robustness and complex data structures. Springer, Heidelberg (2013) 17–34. | MR

K. Mosler and Y. Polyakova, General notions of depth for functional data. arXiv:1208.1981 (2012).

S. Nagy, Coordinatewise characteristics of functional data. In Proc. 31th Int. Conf. Mathematical Methods in Economics 2013, Jihlava, Czech Republic, edited by H. Vojcáˇková. (Part II). College of Polytechnics Jihlava, September (2013) 655–660.

J.O. Ramsay and B.W. Silverman, Functional data analysis. Springer Series in Statistics, 2nd edition. Springer, New York (2005). | MR | Zbl

A.W. Roberts and D.E. Varberg, Convex functions. Vol. 57 of Pure and Applied Mathematics. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1973). | MR | Zbl

R. Serfling, Depth functions in nonparametric multivariate inference. In Data depth: robust multivariate analysis, computational geometry and applications. Vol. 72 of DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc., Providence, RI (2006) 1–16. | MR | Zbl

R. Serfling, Multivariate symmetry and asymmetry. Vol. 8 of Encyclopedia of Statistical Sciences, 2nd edition (2006) 5338–5345.

J.W. Tukey, Mathematics and the picturing of data. In Proc. of the International Congress of Mathematicians (Vancouver, B. C., 1974). Vol. 2, Canad. Math. Congress, Montreal, Que. (1975) 523–531. | MR | Zbl

A.W. van der Vaart and J.A. Wellner, Weak convergence and empirical processes. Springer Series in Statistics. Springer-Verlag, New York (1996). | MR | Zbl

Y. Zuo and R. Serfling, General notions of statistical depth function. Ann. Statist. 28 (2000) 461–482. | MR | Zbl

Y. Zuo and R. Serfling, On the performance of some robust nonparametric location measures relative to a general notion of multivariate symmetry. J. Statist. Plann. Inference 84 (2000) 55–79. | DOI | MR | Zbl