×

Evaluating the complexity of some families of functional data. (English) Zbl 1395.62085

Summary: In this paper we study the complexity of a functional data set drawn from particular processes by means of a two-step approach. The first step considers a new graphical tool for assessing to which family the data belong: the main aim is to detect whether a sample comes from a monomial or an exponential family. This first tool is based on a nonparametric kNN estimation of small ball probability. Once the family is specified, the second step consists in evaluating the extent of complexity by estimating some specific indexes related to the assigned family. It turns out that the developed methodology is fully free from assumptions on model, distribution as well as dominating measure. Computational issues are carried out by means of simulations and finally the method is applied to analyse some financial real curves dataset.

MSC:

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aneiros, G., Bongiorno, E.G., Cao, R. and Vieu, P. (2017). Functional Statistics and Related Fields. Springer. · Zbl 1373.62016
[2] Biau, G., C´erou, F. and Guyader, A. (2010). Rates of convergence of the functional k-nearest neighbor estimate. IEEE Transactions on Information Theory, 56, 2034-2040. · Zbl 1366.62080
[3] Biau, G. and Devroye, L. (2015). Lectures on the Nearest Neighbor Method. Springer Series in the Data Sciences. Springer, Cham. · Zbl 1330.68001
[4] Bogachev, V.I. (1998). Gaussian Measures. Vol. 62 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI.
[5] Bongiorno, E.G. and Goia, A. (2016). Classification methods for Hilbert data based on surrogate density. Computational Statistics & Data Analysis, 99, 204-222. · Zbl 1468.62030
[6] Bongiorno, E.G. and Goia, A. (2017). Some insights about the small ball probability factorization for Hilbert random elements. Statistica Sinica, Forthcoming. · Zbl 1392.60008
[7] Bongiorno, E.G., Goia, A., Salinelli, E. and Vieu, P. (Eds.) (2014). Contributions in Infinite-Dimensional Statistics and Related Topics. Societ‘a Editrice Esculapio. · Zbl 1377.62023
[8] Bosq, D. (2000). Linear Processes in Function Spaces. Vol. 149 of Lecture Notes in Statistics. SpringerVerlag, New York. · Zbl 0962.60004
[9] Burba, F., Ferraty, F. and Vieu, P. (2009). k-nearest neighbour method in functional nonparametric regression. Journal of Nonparametric Statistics, 21, 453-469. · Zbl 1161.62017
[10] Campbell, J.Y., Lo, A.W.-C. and MacKinlay, A.C. (1997). The Econometrics of Financial Markets. Princeton University Press. · Zbl 0927.62113
[11] Cardot, H., C´enac, P. and Godichon-Baggioni, A. (2017). Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls. Annals of Statistics, 45, 591-614. · Zbl 1371.62027
[12] Chen, K., Delicado, P. and M¨uller, H.-G. (2017). Modelling function-valued stochastic processes, with applications to fertility dynamics. Journal of the Royal Statistical Society. Series B, Statistical Methodology, 79, 177-196.
[13] Ciollaro, M., Genovese, C., Lei, J. and Wasserman, L. (2014). The Functional Mean-Shift Algorithm for Mode Hunting and Clustering in Infinite Dimensions. Preprint.
[14] Delaigle, A. and Hall, P. (2010). Defining probability density for a distribution of random functions. Annals · Zbl 1183.62061
[15] 44Evaluating the complexity of some families of functional data
[16] Gy¨orfi, L., Kohler, M., Krzyzak, A.and Walk, H. (2006). A Distribution-Free Theory of Non-Parametric Regression. Springer Science & Business Media.
[17] H¨ardle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Annals of Statistics, 21, 1926-1947. · Zbl 0795.62036
[18] Horv´ath, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. Springer, New York.
[19] Jacques, J. and Preda, C. (2014). Functional data clustering: a survey. Advances in Data Analysis and Classification, 8, 231-255.
[20] Kara, L.-Z., Laksaci, A., Rachdi, M. and Vieu, P. (2017). Data-driven kNN estimation in nonparametric functional data analysis. Journal of Multivariate Analysis, 153, 176-188. · Zbl 1351.62084
[21] Kloeden, P.E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Vol. 23 of Applications of Mathematics (New York). Springer-Verlag, Berlin. · Zbl 0752.60043
[22] Kokoszka, P., Oja, H., Park, B. and Sangalli, L. (2017). Special issue on functional data analysis. Econo- metrics and Statistics, 1, 99-100.
[23] Kudraszow, N.L. and Vieu, P. (2013). Uniform consistency of kNN regressors for functional variables. Statistics & Probability Letters, 83, 1863-1870. · Zbl 1277.62113
[24] Lalo¨e, T. (2008). A k-nearest neighbor approach for functional regression. Statistics & Probability Letters, 78, 1189-1193. · Zbl 1140.62033
[25] Li, W.V., Shao and Q.-M. (2001). Gaussian processes: inequalities, small ball probabilities and applications. In: Stochastic Processes: Theory and Methods. Vol. 19 of Handbook of Statistics NorthHolland, Amsterdam, pp. 533-597. · Zbl 0987.60053
[26] Lian, H. (2011). Convergence of functional k-nearest neighbor regression estimate with functional responses. Electronic Journal of Statistics, 5, 31-40. · Zbl 1274.62291
[27] Lifshits, M.A. (2012). Lectures on Gaussian Processes. Springer Briefs in Mathematics. Springer, Heidelberg. · Zbl 1248.60002
[28] Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality. Stochastic Processes and their Applications, 115, 155-177. · Zbl 1101.62031
[29] Nikitin, Y.Y. and Pusev, R.S. (2013). Exact small deviation asymptotics for some Brownian functionals. Theory of Probability and Its Applications, 57, 60-81. · Zbl 1278.60072
[30] Ramsay, J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.