Local linear estimation of the conditional mode under left truncation for functional regressors

Boudada, Halima ; Leulmi, Sarra

Kybernetika, Tome 59 (2023) no. 4, pp. 548-574

Abstract (VO)

In this work, we introduce a local linear estimator of the conditional mode for a random real response variable which is subject to left-truncation by another random variable where the covariate takes values in an infinite dimensional space. We first establish both of pointwise and uniform almost sure convergences, with rates, of the conditional density estimator. Then, we deduce the strong consistency of the obtained conditional mode estimator. We finally illustrate the outperformance of our method with respect to the kernel one through a simulation study for a finite sample with different rates of truncation and sizes.

Détail

DOI : 10.14736/kyb-2023-4-0548

Classification : 62G07, 62G20, 62N99, 62R10
Keywords: functional regressors; left truncation model; conditional mode; almost sure convergence; local linear estimator

Bibliographie

[1] Hennania, L. Ait, Lemdania, M., Said, E. Ould: Robust regression analysis for a censored response and functional regressors. J. Nonparametr. Statist. 31 (2019), 221-2430. | DOI | MR

[2] Barrientos-Marin, J., Ferraty, F., Vieu, P.: Locally modelled regression and functional data. J.Nonparametr. Statist. 22 (2010), 617-632. | DOI | MR

[3] Benkhaled, A., Madani, F., Khardani, S.: Strong consistency of local linear estimation of a conditional density function under random censorship. Arab. J. Math. 9 (2020), 513-529. | DOI | MR

[4] Boudada, H.: A nonparametric estimation of the conditional quantile for truncated and functional data. Int. J. Math. Oper. Res. 21 (2022), 127-140. | DOI | MR

[5] Boudada, H., Leulmi, S., Kharfouch, S.: Rate of the almost sure convergence of a generalized regression estimate based on truncated and functional data. J. Sib. Fed. Univ. - Math. Phys. 13 (2020), 480-491. | DOI | MR

[6] Demongeot, J., Laksaci, A., Madani, F., Rachdi, M.: Functional data analysis: conditional density estimation and its application. Statist. 47 (2013), 26-44. | DOI | MR

[7] Derrar, S., Laksaci, A., Sa\"{i}d, E. Ould: On the nonparametric estimation of the functional $\psi$-regression for a random left-truncation model. J. Stat. Theory Pract. 9 (2015), 823-849. | DOI | MR

[8] Derrar, S., Laksaci, A., Sa\"{i}d, E. Ould: M-estimation of the regression function under random left truncation and functional time series model. Statist. Papers 61 (2018), 1181-1202. | DOI | MR

[9] Fan, J.: Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 (1992), 998-1004. | DOI | MR

[10] Ferraty, F., Laksaci, A., Tadj, A., Vieu, P.: Rate of uniform consistency for nonparametric estimates with functional variables. J. Statist. Plan, Inference 140 (2010), 335-352. | DOI | MR

[11] Ferraty, F., Vieu, P.: Nonparametric Functional Data Analysis. Theory and Practice. Springer Ser. Statist. New York 2006. | MR

[12] He, S., Yang, G. L.: Estimation of the truncation probability in the random truncation model. Ann. Statist. 26 (1998), 1011-1027. | DOI | MR

[13] Helal, N., Ould-Sa\"{i}d, E.: Kernel conditional quantile estimator under left truncation for functional regressors. Opuscula Math. 36 (2016), 25-48. | DOI | MR

[14] Horrigue, W., Sa\"{i}d, E. Ould: Strong uniform consistency of a nonparametric estimator of a conditional quantile for censored dependent data and functional regressors. Random Oper.Stoch. Equat. 19 (2011), 131-156. | DOI | MR

[15] Lemdani, M., Ould-Sa\"{i}d, E.: Asymptotic behavior of the hazard rate kernel estimator under truncated and censored data. Comm. Statist. Theory Methods. 36 (2007), 155-173. | DOI | MR

[16] Leulmi, S.: Local linear estimation of the conditional quantile for censored data and functional regressors. Comm. Statist. Theory Methods 50 (2019), 3286-3300. | DOI | MR

[17] Leulmi, S.: Nonparametric local linear regression estimation for censored data and functional regressors. J. Korean Statist. Soc. (2020), 25-46. | DOI | MR

[18] Leulmi, S., Messaci, F.: Local linear estimation of a generalized regression function with functional dependent data. Comm. Statist. Theory Methods 47 (2018), 5795-5811. | DOI | MR

[19] Leulmi, S., Messaci, F.: A Class of Local Linear Estimators with Functional Data. J. Sib. Fed. Univ. - Math. Phys. 12 (2019), 379-391. | DOI | MR

[20] Mechab, B., Hamidi, N., Benaissa, S.: Nonparametric estimation of the relative error in functional regression and censored data. Chil. J. Statist. 10 (2019), 177-195. | MR

[21] Messaci, F., Nemouchi, N., Ouassou, I., Rachdi, M.: Local polynomial modelling of the conditional quantile for functional data. Stat. Methods Appl. 24 (2015), 597-622. | DOI | MR

[22] Stute, W.: Almost sure representations of the product-limit estimator for truncated data. Ann. Statist. 21 (1993), 146-156. | DOI | MR

[23] Woodroofe, M.: Estimating a distribution function with truncated data. Ann. Statist. 13 (1985), 163-177. | DOI | MR