Strong uniform consistency rates of conditional quantiles for time series data in the single functional index model

Abstract

The main objective of this paper is to estimate non-parametrically the quantiles of a conditional distribution when the sampleis considered as anα-mixing sequence. First of all, a kernel type estimator for the conditional cumulative distribution function (condcdf ) is introduced. Afterwards, we give an estimation of the quantiles by inverting this estimated cond-cdf, the asymptotic propertiesare stated when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniformalmost complete convergence (with rate) of the kernel estimate of this model are established. This approach can be applied in timeseries analysis. For that, the whole observed time series has to be split into a set of functional data, and the functional conditionalquantile approach can be employed both in foreseeing and building confidence prediction bands

Keywords

• Conditional quantile,conditional cumulative distribution,derivatives of conditional cumulative distribution,functionalrandom variable,kernel estimator,nonparametric estimation,strong mixing processes

Strong uniform consistency rates of conditional quantiles for time series data in the single functional index model

Öz

Kaynakça

1. Aneiros P ´erez, G., Cardot, H., Est ´evez-P ´erez, G., Vieu, P.Maximum ozone forecasting by functional nonparametric. approach. Environmetrics. Vol. 15 (2004), pp. 675-685.
2. Benhenni, K., Hedli-Griche, S., Rachdi M., Vieu, P. Consistency of the regression estimator with functional data under long memory conditions. J. of Statist. Probab. Lett. Vol. 78 no. (8) (2008), pp. 10431049.
3. Besse, P., Cardot, H., Stephenson, D. Autoregressive forecasting of some functional climatic variations. Scand. J. of Statist. Vol. 27 (2000), pp. 673-687.
4. ] Bouchentouf, A. A., Djebbouri, T., Rabhi, A., Sabri, K. Strong uniform consistency rates of some characteristics of the conditional distribution estimator in the functional single-index model. Appl. Math. (Warsaw). Vol. 41 no. 4 (2014), pp. 301-322.
5. Bosq, D.Linear processs in function spaces. Lecture Notes in Statistics. Vol. 149, SpringerVerlag (2000).
6. Bosq, D., Lecoutre, J. P.Th ´eorie de l’estimation fonctionnelle, ECONOMICA, Paris, 1987.
7. Cai, Z.Regression quantiles for time series. Econometric Theory. Vol. 18 (2002), pp. 169-192.
8. Dabo-Niang, S., Ferraty, F., Vieu, P.Nonparametric unsupervised classification of satellite wave altimeter forms. Computational Statistics (2004), Ed. J. Antoch. Physica Verlag, Berlin. pp. 879-887.

9. Damon, J., Guillas, S.The inclusion of exogenous variables in functional autoregressive ozone forecasting. Environmetrics. Vol. 13 (2002), pp. 759-774.
10. Fern ´andez de Castro, B., Guillas, S., Gonz ´alez Manteiga, W. Functional samples and bootstrap for the prediction of SO2 levels. Preprint. (2003).
11. Ferraty, F., Goia, A., Vieu, P. Functional nonparametric model for times series: A fractal approach for dimensional reduction. TEST. Vol. 11 no. 2 (2002), pp. 317-344.
12. Ferraty, F., Laksaci, A., Tadj, A., Vieu, P. Rate of uniform consistency for nonparametric estimates with functional variables. J. Statist. Plann. and Inf. Vol. 140 (2010), pp. 335-352.
13. Ferraty, F., Rabhi, A., Vieu, P.Conditional quantiles for functional dependent data with application to the climatic El Nin ˜o phenomenon, Sankhy ˜a The Indian Journal of Statistics, Special Issue on Quantile Regression and Related Methods, Vol. 67 no. 2 (2005), pp. 378-399.
14. Ferraty, F., Vieu, P.The functional nonparametric model and application to spectrometric data. Computational Statistics. Vol. 17 no. 4 (2002), pp. 545-564.
15. Ferraty, F., Vieu, P.Curves discrimination: a nonparametric functional approach. Computational Statistics and Data Analysis. Vol. 44 (2003), pp. 161-173.
16. Ferraty, F., Vieu, P.Functional nonparametric statistics: a double infinite dimensional framework. Recent advanvces and trends in Nonparametric Statistics, Ed. M. Akritas and D. Politis, Elsevier, (2003b).
17. Ferraty, F., Vieu, P.Nonparametric methods for functional data, with applications in regression, time-series prediction and curves discrimination. J. Nonparametr. Stat. Vol. 16 (2004), pp. 11-126.
18. Ferraty, F., Vieu, P.Nonparametric Functional Data Analysis: Theory and Practice, Springer Series in Statistics, Springer, New York, 2006.
19. Gannoun, A., Saracco, J., Yu, K. Nonparametric prediction by conditional median and quantiles. J. Statist. Plann. Inference. Vol. 117 (2003), pp. 207-223.
20. Gasser, T., Hall, P., Presnell, B. Nonparametric estimation of the mode of a distribution of random curves. J. R. Statist. Soc, B. Vol. 60 no 4 (1998), pp. 681-691.
21. Hall, P., Poskitt, P. and Presnell, D.A functional data-analytic approach to signal discrimination. Technometrics. Vol. 35 (2001), pp. 140-143.
22. Hall, P., Heckman, N.Estimating and depicting the structure of the distribution of random functions. Biometrika. Vol. 89 (2002), pp. 145-158.
23. Ramsay, J. O., Silverman, B. W. Functional data analysis. Springer Series in Statistics (1997).
24. Ramsay, J. O., Silverman, B. W. Applied functional data analysis. Springer-Verlag, (2002).
25. Rio, E. Th ´eorie asymptotique des processus al ´eatoires faiblements d ´ependant, Math ´ematiques Application, Vol. 22 no. 4 (2000), pp. 331-334.
26. Roussas, G. Nonparametric estimation of the transition distribution function of a Markov process. Ann. Math. Statist. Vol. 40 (1969), pp. 1386-1400.
27. Samanta, M. Nonparametric estimation of conditional quantiles. Statist. Proba. Letters. Vol. 7 (1989), pp. 407-412.
28. Wang, H., Zhao, Y. A kernel estimator for conditional t-quantiles for mixing samples and its strong uniform convergence, (in chinese). J. Math. Appl. (Wuhan). Vol. 12 (1999), pp. 123-127.
29. Zhou, Y., Liang, H. Asymptotic properties for L1norm kernel estimator of conditional median under dependence. J. Nonparametr. Stat. Vol. 15 (2003), pp. 205-219.