Kybernetika 59 no. 4, 548-574, 2023

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

Halima Boudada and Sarra LeulmiDOI: 10.14736/kyb-2023-4-0548

Abstract:

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.

Keywords:

local linear estimator, functional regressors, left truncation model, conditional mode, almost sure convergence

Classification:

62G07, 62G20, 62R10, 62N99

paper.pdf

References:

  1. L. Ait Hennania, M. Lemdania and E. Ould Said: Robust regression analysis for a censored response and functional regressors. J. Nonparametr. Statist. 31 (2019), 221-2430.   DOI:10.1080/10485252.2018.1546386
  2. J. Barrientos-Marin, F. Ferraty and P. Vieu: Locally modelled regression and functional data. J.Nonparametr. Statist. 22 (2010), 617-632.   DOI:10.1080/10485250903089930
  3. A. Benkhaled, F. Madani and S. Khardani: Strong consistency of local linear estimation of a conditional density function under random censorship. Arab. J. Math. 9 (2020), 513-529.   DOI:10.1007/s40065-020-00282-1
  4. H. Boudada: A nonparametric estimation of the conditional quantile for truncated and functional data. Int. J. Math. Oper. Res. 21 (2022), 127-140.   DOI:10.1504/ijmor.2022.120318
  5. H. Boudada, S. Leulmi and S. Kharfouch: 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:10.17516/1997-1397-2020-13-4-480-491
  6. J. Demongeot, A. Laksaci, F. Madani and M. Rachdi: Functional data analysis: conditional density estimation and its application. Statist. 47 (2013), 26-44.   DOI:10.1080/02331888.2011.568117
  7. S. Derrar, A. Laksaci and E. Ould Sa\"{i}d: On the nonparametric estimation of the functional $\psi$-regression for a random left-truncation model. J. Stat. Theory Pract. 9 (2015), 823-849.   DOI:10.1080/15598608.2015.1032455
  8. S. Derrar, A. Laksaci and E. Ould Sa\"{i}d: M-estimation of the regression function under random left truncation and functional time series model. Statist. Papers 61 (2018), 1181-1202.   DOI:10.1007/s00362-018-0979-z
  9. J. Fan: Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87 (1992), 998-1004.   DOI:10.1080/01621459.1992.10476255
  10. F. Ferraty, A. Laksaci, A. Tadj and P. Vieu: Rate of uniform consistency for nonparametric estimates with functional variables. J. Statist. Plan, Inference 140 (2010), 335-352.   DOI:10.1016/j.jspi.2009.07.019
  11. F. Ferraty and P. Vieu: Nonparametric Functional Data Analysis. Theory and Practice. Springer Ser. Statist. New York 2006.   CrossRef
  12. S. He and G. L. Yang: Estimation of the truncation probability in the random truncation model. Ann. Statist. 26 (1998), 1011-1027.   DOI:10.1214/aos/1024691086
  13. N. Helal and E. Ould-Sa\"{i}d: Kernel conditional quantile estimator under left truncation for functional regressors. Opuscula Math. 36 (2016), 25-48.   DOI:10.7494/OpMath.2016.36.1.25
  14. W. Horrigue and E. Ould Sa\"{i}d: 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:10.1515/ROSE.2011.008
  15. M. Lemdani and E. Ould-Sa\"{i}d: Asymptotic behavior of the hazard rate kernel estimator under truncated and censored data. Comm. Statist. Theory Methods. 36 (2007), 155-173.   DOI:10.1080/03610920600966506
  16. S. Leulmi: Local linear estimation of the conditional quantile for censored data and functional regressors. Comm. Statist. Theory Methods 50 (2019), 3286-3300.   DOI:10.1080/03610926.2019.1692033
  17. S. Leulmi: Nonparametric local linear regression estimation for censored data and functional regressors. J. Korean Statist. Soc. (2020), 25-46.   DOI:10.1007/s42952-020-00080-7
  18. S. Leulmi and F. Messaci: Local linear estimation of a generalized regression function with functional dependent data. Comm. Statist. Theory Methods 47 (2018), 5795-5811.   DOI:10.1080/03610926.2017.1402048
  19. S. Leulmi and F. Messaci: A Class of Local Linear Estimators with Functional Data. J. Sib. Fed. Univ. - Math. Phys. 12 (2019), 379-391.   DOI:10.17516/1997-1397-2019-12-3-379-391
  20. B. Mechab, N. Hamidi and S. Benaissa: Nonparametric estimation of the relative error in functional regression and censored data. Chil. J. Statist. 10 (2019), 177-195.   CrossRef
  21. F. Messaci, N. Nemouchi, I. Ouassou and M. Rachdi: Local polynomial modelling of the conditional quantile for functional data. Stat. Methods Appl. 24 (2015), 597-622.   DOI:10.1007/s10260-015-0296-9
  22. W. Stute: Almost sure representations of the product-limit estimator for truncated data. Ann. Statist. 21 (1993), 146-156.   DOI:10.1214/aos/1176349019
  23. M. Woodroofe: Estimating a distribution function with truncated data. Ann. Statist. 13 (1985), 163-177.   DOI:10.1214/aos/1176346584