Strong uniform consistency rates of conditional density estimation in the single functional index model for functional data under random censorship. (English) Zbl 1491.62028
Summary: The main objective of this paper is to investigate the estimation of conditional density function based on the single-index model in the censorship model when the sample is considered as an independent and identically distributed (i.i.d.) random variables. First of all, a kernel type estimator for the conditional density function (cond-df) is introduced. Afterwards, the asymptotic properties are stated when the observations are linked with a single-index structure. The pointwise almost complete convergence and the uniform almost complete convergence (with rate) of the kernel estimate of this model are established. As an application the conditional mode in functional single-index model is presented. Finally, a simulation study is carried out to evaluate the performance of this estimate.
MSC:
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62R10 | Functional data analysis |
Keywords:
conditional density; functional single-index process; functional random variable; nonparametric estimation; small ball probabilityReferences:
| [1] | Aït-Saidi, A.; Ferraty, F. and Kassa, R. (2005). Single functional index model for a time series, Revue Roumaine de Mathématique Pures et Appliquées, 50, 321-330. · Zbl 1084.62035 |
| [2] | Aït-Saidi, A.; Ferraty, F.; Kassa, R. and Vieu, P. (2008). Cross-validated estimation in the single functional index model, Statistics, 24, 475-494. · Zbl 1274.62519 |
| [3] | Attaoui, S. (2014). On the nonparametric conditional density and mode estimates in the single functional index model with strongly mixing data, Sankhyã: The Indian Journal of Statistics, 76(A), 356-378. · Zbl 1307.62089 |
| [4] | Attaoui, S. (2014). Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data, J. AStA., 98, 257-286. · Zbl 1443.62080 |
| [5] | Attaoui, S.; Laksaci, A. and Ould-Saïd, E. (2011). A note on the conditional density estimate in the single functional index model, Statist. Probab. Lett., 81, 45-53. · Zbl 1206.62058 |
| [6] | Attaoui, S. and Ling, N. (2016). Asymptotic results of a nonparametric conditional cumu-lative distribution estimator in the single functional index modeling for time series data with applications, Metrika, 79, 485-511. · Zbl 1351.62061 |
| [7] | Bouchentouf, A.A.; Djebbouri, T.; Rabhi, A. and Sabri, K. (2014). Strong uniform consistency rates of some characteristics of the conditional distribution estimator in the func-tional single-index model, Appl. Math. (Warsaw), 41(4), 301-322. · Zbl 1308.62058 |
| [8] | Cuevas, A. (2014). A partial overview of the theory of statistics with functional data, J. Statist. Plann. Inference., 147, 1-23. · Zbl 1278.62012 |
| [9] | Deheuvels, P. and Einmahl, J.H.J. (2000). Functional limit laws for the increments of Kaplan-Meier product-limit processes and applications, Ann. Probab., 28(3), 1301-1335. · Zbl 1016.62031 |
| [10] | Ezzahrioui, M. and Ould Saïd, E. (2008). Asymptotic results of a nonparametric con-ditional quantile estimator for functional time series, Comm. Statist. Theory and Methods, 37(16-17), 2735-2759. · Zbl 1147.62030 |
| [11] | Ezzahrioui, M. and Ould Saïd, E. (2010). Some asymptotic results of a nonparametric conditional mode estimator for functional time series data, Statist. Neerlandica, 64, 171-201. · Zbl 07881742 |
| [12] | Ferraty, F.; Laksaci, A.; Tadj, A. and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables, J. Statist. Plann. and Inf., 140, 335-352. · Zbl 1177.62044 |
| [13] | Ferraty, F.; Laksaci, A. and Vieu, P. (2006). Estimating some characteristics of the conditional distribution in nonparametric functional models, Statist. Inference Stoch. Process, 9, 47-76. · Zbl 1117.62030 |
| [14] | Ferraty, F.; Peuch, A. and Vieu, P. (2003). Modèleà indice fonctionnel simple, C. R. Acad. Sci., Paris, 336, 1025-1028. · Zbl 1020.62032 |
| [15] | Ferraty, F. and Vieu, P. (2003). Functional nonparametric statistics: a double infinite dimensional framework. In “Recent Advances and Trends in Nonparametric Statistics” (M. Akritas and D. Politis, Ed.), Elsevier. · Zbl 1429.62241 |
| [16] | Ferraty, F. and Vieu, P. (2006). Nonparametric Functional Data Analysis: Theory and Practice, Springer Series in Statistics, Springer, New York. · Zbl 1119.62046 |
| [17] | Goia, A. and Vieu, P. (2015). A partitioned single functional index model, Computational Statistics, 30(3), 673-692. · Zbl 1342.65034 |
| [18] | Goia, A. and Vieu, P. (2016). An introduction to recent advances in high/infinite dimen-sional statistics, J. Mutiv. Anal., 146, 1-6. · Zbl 1384.00073 |
| [19] | Härdle, W. and Marron, J.S. (1985). Optimal bandwidth selection in nonparametric regression function estimation, Ann. Statist., 13, 1465-1481. · Zbl 0594.62043 |
| [20] | Hristache, M.; Juditsky, A. and Spokoiny, V. (2001). Direct estimation of the index coefficient in the single-index model, Ann. Statist., 29, 595-623. · Zbl 1012.62043 |
| [21] | Hsing, T. and Eubank, R. (2015). Theoretical Foundations of Functional Data Analysis, With an Introduction to Linear Operators, Wiley, Chichester. · Zbl 1338.62009 |
| [22] | Khardani, S.; Lemdani, M. and Ould Saïd, E. (2010). Some asymptotic properties for a smooth kernel estimator of the conditional mode under random censorship, Journal of the Korean Statistical Society, 39, 455-469. · Zbl 1294.62105 |
| [23] | Khardani, S.; Lemdani, M. and Ould Saïd, E. (2011). Uniform rate of strong consistency for a smooth Kernel estimator of the conditional mode under random censorship, J. Statist. Plann. and Inf., 141, 3426-3436. · Zbl 1221.62063 |
| [24] | Khardani, S.; Lemdani, M. and Ould Saïd, E. (2014). On the Central Limit Theorem for a conditional mode estimator of a randomly censored time series, Journal of Statistical Theory and Practice, 8, 722-742. |
| [25] | Ling, N. and Xu, Q. (2012). Asymptotic normality of conditional density estimation in the single index model for functional time series data, Statistics and Probability Letters, 82, 2235-2243. · Zbl 1471.62470 |
| [26] | Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality, Stochastic Process. Appl., 115, 155-177. · Zbl 1101.62031 |
| [27] | Mehra, K.L.; Ramakrishnaiah, Y.S. and Sashikala, P. (2000). Laws of iterated log-arithm and related asymptotics for estimators of conditional density and mode, Ann. Inst. Math. Stat., 52, 630-645. · Zbl 0978.62042 |
| [28] | Ould-Saïd, E. (2006). A strong uniform convergence rate of Kernel conditional quantile estimator under random censorship, Statistics and Probability Letters, 76, 579-586. · Zbl 1141.62319 |
| [29] | Ould-Saïd, E. and Cai, Z. (2005). Strong uniform consistency of nonparametric estimation of the censored conditional mode function, Nonparametric Statistics, 17(7), 797-806. · Zbl 1079.62051 |
| [30] | Ramsay, J. and Silverman, B. (2002). Applied Functional Data Analysis: Methods and Case Studies, Springer, New York. · Zbl 1011.62002 |
| [31] | Ramsay, J. and Silverman, B. (2005). Functional Data Analysis (2nd ed.), Springer, New York. · Zbl 1079.62006 |
| [32] | Samanta, M. and Thavaneswaran, A. (1990). Nonparametric estimation of the conditional mode, Commun. Stat. Theory Methods, 16, 4515-4524. · Zbl 0732.62037 |
| [33] | Xia, Y. and Härdle, W. (2006). semi-parametric estimation of partially linear single-index models, J. Multivar. Anal., 97, 1162-1184. · Zbl 1089.62050 |
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