Missing at random in nonparametric regression for functional stationary ergodic data in the functional index model
The main objective of this paper is to estimate non-parametrically the the estimator for the regression function operator when the observations are linked with a single-index. The functional stationary ergodic data with missing at random (MAR) are considered.In particular, we construct the kernel type estimator of the regression operator, some asymptotic properties such as the convergence rate in probability as well as the asymptotic normality of the estimator are established under some mild conditions respectively. As an application, the asymptotic $(1 -\zeta)$ confidence interval of the regression operator is also presented for $0 < \zeta < 1.$
M. Lothaire, Combinatorics on words, Addison-Wesley, Reading, MA, 1983.
I. A. Ahmad, Uniform strong convergence of the generalized failure rate estimate, Bull. Math. Statist., 17 (1976), 77–84.
D. Bosq, J. P. Lecoutre, Th´eorie de l’estimation fonctionnelle, ECONOMICA (eds), Paris, 1987.
H. Cardot, F. Ferraty, P. Sarda, Functional linear model, Statist. Probab. Lett., 45 (1999), 11–22.
G. Aneiros P´erez, H. Cardot, G. Est´evez-P´erez and P. Vieu, Maximum ozone forecasting by functional nonparametric approach, Environmetrics, 15 (2004), 675–685.
K . Benhenni, S. Hedli-Griche, M. Rachdi and P. Vieu, Consistency of the regression estimator with functional data under long memory conditions, J. of Statist. Probab. Lett., 78 no. (8) (2008), 1043–10 ASYMPTOTIC PROPERTIES FOR REGRESSION FUNCTION WITH MISSING AT RANDOM13
P. Besse, H. Cardot and D. Stephenson, Autoregressive forecasting of some functional climatic variations, Scand. J. of Statist., 27 (2000), 673–687.
A.A. Bouchentouf, T. Djebbouri, R. Rabhi and K. Sabri, Strong uniform consistency rates of some characteristics of the conditional distribution estimator in the functional single-index model, Appl. Math. (Warsaw), 41 no. 4 (2014), 301–322.
D. Bosq, Linear processs in function spaces. Lecture Notes in Statistics. 149, Springer-Verlag, 2000.
D. Bosq and J.P. Lecoutre, Th´eorie de l’estimation fonctionnelle. ECONOMICA, Paris, 1987.
Cai, Z. Regression quantiles for time series. Econometric Theory. Vol. 18 (2002), pp. 169-192.
P. Chaudhuri, K. Doksum and A. Samarov, On average derivative quantile regression, Ann. Statist., 25 (1997), 715–744.
S. Dabo-Niang, F. Ferraty and P. Vieu, Nonparametric unsupervised classification of satellite wave altimeter forms, Computational Statistics. 2004, Ed. J. Antoch. Physica Verlag, Berlin. 879–887.
J. Damon and S. Guillas, The inclusion of exogenous variables in functional autoregressive ozone forecasting, Environmetrics, 13 (2002), 759–774.
B. Fern´andez de Castro, S. Guillas and W. Gonz´alez Manteiga, Functional samples and bootstrap for predicting sulfur dioxide levels, Technometrics, 47 no. 2 (2005), 212–222.
F. Ferraty, A. Goia and P. Vieu, Functional nonparametric model for times series: A fractal approach for dimensional reduction, TEST, 11 no. 2 (2002), 317–344.
F. Ferraty, A. Laksaci, A. Tadj and P. Vieu, Rate of uniform consistency for nonparametric estimates with functional variables, J. Statist. Plann. and Inf., 140 (2010), 335–352.
F. Ferraty, A. Rabhi and P. Vieu, Conditional quantiles for functional dependent data with application to the climatic ElNin˜o phenomenon, Sankhy˜a: The Indian Journal of Statistics, Special Issue on Quantile Regression and Related Methods, 67 no. 2 (2005), 378–399.
F. Ferraty and P. Vieu, The functional nonparametric model and application to spectrometric data, Computational Statistics, 17 no. 4 (2002), 545–564.
Ferraty, F., Vieu, P. Curves discrimination: a nonparametric functional approach. Computational Statistics and Data Analysis. Vol. 44 (2003), pp. 161-173.
F. Ferraty and P. Vieu, Functional nonparametric statistics: a double infinite dimensional framework. Recent advanvces and trends in Nonparametric Statistics, Ed. M. Akritas and D. Politis, Elsevier, 2003b.
F. Ferraty and P. Vieu, Nonparametric methods for functional data, with applications in regression, time-series prediction and curves discrimination, J. Nonparametr. Stat., 16(2004), 111–126.
F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Theory and Practice. Springer Series in Statistics, Springer, New York, 2006.
A. Gannoun, J. Saracco and K. Yu, Nonparametric prediction by conditional median and quantiles, J. Statist. Plann. Inference., 117 (2003), 207–223.
T. Gasser, P. Hall and B. Presnell, Nonparametric estimation of the mode of a distribution of random curves, J. R. Statist. Soc., B., 60 no 4 (1998), 681–691.
P. Hall, P. Poskitt and D. Presnell, A functional data-analytic approach to signal discrimination, Technometrics, 35 (2001), 140–143.
P. Hall and N. Heckman, Estimating and depicting the structure of the distribution of random functions, Biometrika, 89 (2002), 145–158.
J.P. Lecoutre and E. Ould-Sa¨ıd, Hazard rate estimation for strong mixing and censored processes, J. Nonparam. Statist., 5 (1995), 83–89.
W.J. Padgett, Nonparametric estimation of density and hazard rate functions when samples are censored. In P.R. Krishnaiah and C.R. Rao (Eds), Handbook of Statistics, 7, 313–331. Elsevier Science Publishers, 1988.
J.O. Ramsay and B.W. Silverman, Functional data analysis. Springer Series in Statistics 1997.
J.O. Ramsay and B.W. Silverman, Applied functional data analysis. Springer-Verlag, 2002.
E. Rio, Th´eorie asymptotique des processus al´eatoires faiblements d´ependant, Math´ematiques Application, 22 no. 4 (2000), 331–334.
G. Roussas, Nonparametric estimation of the transition distribution function of a Markov process, Ann. Math. Statist., 40 (1969), 1386–1400. 14 F. AKKAL ET AL.
M. Samanta, Nonparametric estimation of conditional quantiles, Statist. Proba. Letters., 7 (1989), 407–412.
M. Tanner and W.H. Wong, The estimation of the hazard function from randomly censored data by the kernel methods, Ann. Statist., 11 (1983), 989–993.
Van Keilegom, I. and Veraverbeke, N. Hazard rate estimation in nonparametric regression with censored data. Ann. Inst. Statist. Math. Vol. 53 (2001), pp. 730-745.
H. Wang and Y. Zhao, A kernel estimator for conditional t-quantiles for mixing samples and its strong uniform convergence, (in chinese), J. Math. Appl. (Wuhan), 12 (1999), 123–127.
Y. Zhou and H. Liang, Asymptotic properties for L1 norm kernel estimator of conditional median under dependence, J. Nonparametr. Stat., 15 (2003), 205–219. Statistics Laboratory Stochastic Processes, University Djillali LIABES of Sidi Bel Abbes, Algeria. E-mail address: firstname.lastname@example.org Faculty of Exact Sciences Department Maths and Computer Science, Tahri Mohamed University of Bechar, Algeria. E-mail address: email@example.com Laboratory of Mathematics, University Djillali LIABES of Sidi Bel Abbes, Algeria. E-mail address: rabhi firstname.lastname@example.org
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