Strong consistency of the local linear relative regression estimator for censored data

• Autorzy: Bouhadjera Feriel , Saïd Elias Ould

Identyfikatory

DOI

10.7494/OpMath.2022.42.6.805

• Języki publikacji: EN

Abstrakty

EN

In this paper, we combine the local linear approach to the relative error regression estimation method to build a new estimator of the regression operator when the response variable is subject to random right censoring. We establish the uniform almost sure consistency with rate over a compact set of the proposed estimator. Numerical studies, firstly on simulated data, then on a real data set concerning the death times of kidney transplant patients, were conducted. These practical studies clearly show the superiority of the new estimator compared to competitive estimators.

Słowa kluczowe

EN

censored data; local linear approach; relative error; regression function; uniform almost sure convergence

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2022
• Tom: Vol. 42, no. 6
• Strony: 805--832
• Opis fizyczny: Bibliogr. 31 poz., tab., wykr.

Twórcy

autor

Bouhadjera Feriel

• MISTEA, Université Montpellier, INRAE, Institut Agro, 2 place Pierre Viala, Montpellier, 34060, France

• https://orcid.org/0000-0001-9417-1086

autor

Saïd Elias Ould

• Université du Littoral Côte d’Opale, Laboratoire de Mathématiques Pures et Appliquées, IUT de Calais, 19, rue Louis David, Calais, 62228, France

• https://orcid.org/0000-0002-5068-3140

Bibliografia

• [1] M. Attouch, A. Laksaci, N. Messabihi, Nonparametric relative error regression for spatial random variables, Statist. Papers 58 (2017), 987–1008.
• [2] R. Beran, Nonparametric Regression with Randomly Censored Survival Data, Department of Statistics, University of California, Berkeley, 1981.
• [3] F. Bouhadjera, E. Ould Saïd, M.R. Remita, Strong consistency of the nonparametric local linear regression estimation under censorship model, Comm. Statist. Theory Methods 51 (2022), no. 20, 7056-7072.
• [4] A. Carbonez, L. Gyorfi, E.C. Van Der Meulen, Partitioning estimates of a regression function under random censoring, Statist. Decisions 13 (1995), no. 1, 21-37.
• [5] A. Chahad, L. Ait-Hennani, A. Laksaci, Functional local linear estimate for functional relative-error regression, J. Stat. Theory Pract. 11 (2017), no. 4, 771-789.
• [6] K. Chen, S. Guo, Y. Lin, Z. Ying, Least absolute relative error estimation, J. Amer. Statist. Assoc. 105 (2010), no. 491, 1104–1112.
• [7] D.M. Dąbrowska, Nonparametric regression with censored survival data, Scand. J. Statist. 14 (1987), no. 3, 181–197.
• [8] D.M. Dąbrowska, Uniform consistency of the kernel conditional Kaplan–Meier estimate, Ann. of Statist. 17 (1989), no. 3, 1157–1167.
• [9] P. Deheuvels, J.H.J. Einmahl, Functional limit laws for the increments of Kaplan–Meier product limit processes and applications, Ann Probab. 28 (2000), no. 3, 1301–1335.
• [10] A. El Ghouch, I. Van Keilegom, Local linear quantile regression with dependent censored data, Statist. Sinica 19 (2009), no. 4, 1621–1640.
• [11] A. El Ghouch, I. Van Keilegom, Nonparametric regression with dependent censored data, Scandinavian J. of Statist. 35 (2008), no. 2, 228–247.
• [12] J. Fan, Design adaptative nonparametric regression, J. Amer. Statist. Assoc. 87 (1992), no. 420, 998–1004.
• [13] J. Fan, I. Gijbels, Local Polynomial Modelling and its Applications, Monographs on Statistics and Applied Probability 66, Chapman & Hall/CRC, 1996.
• [14] J. Fan, Q. Yao, Nonlinear Time Series: Nonparametric and Parametric Methods, Springer, New York, 2003.
• [15] E. Giné, A. Guillou, Law of the iterated logarithm for censored data, Ann. of Probab. 27 (1999), no. 4, 2042–2067.
• [16] E. Giné, A. Guillou, On consistency of kernel density estimators for randomly censored data: rates holding uniformly over adaptive intervals, Ann. Inst. H. Poincaré Probab. Statist. 37 (2001), no. 4, 503–522.
• [17] Z. Guessoum, E. Ould Saïd, On nonparametric estimation of the regression function under random censorship model, Statist. Decisions 26 (2008), no. 3, 159-177.
• [18] K. Hirose, H. Masuda, Robust relative error estimation, Entropy 20 (2018), no. 9, Paper no. 632.
• [19] Dh. Hu, Local least product relative error estimation for varying coefficient multiplicative regression model, Acta Math. Appl. Sin. Engl. Ser. 35 (2019), no. 2, 274–286.
• [20] M.C. Jones, H. Park, K.I. Shin, S.K. Vines, S.O. Jeong, Relative error prediction via kernel regression smoothers, J. Statist. Plann. Inference 138 (2008), no. 10, 2887-2898.
• [21] E.L. Kaplan, P. Meier, Nonparametric estimation from incomplete observations, J. Amer. Statist. Assoc. 53 (1958), 458–481.
• [22] S. Khardani, Y. Slaoui, Nonparametric relative regression under random censorship model, Statist. and Probab. Letters 151 (2019), 116–122.
• [23] J.P. Klein, M.L. Moeschberger, Survival Analysis: Techniques for Censored and Truncated Data, Springer-Verlag, New York, 2004.
• [24] M. Kohler, K. Máthé, M. Pintér, Prediction from randomly right censored data, J. Multivariate Anal. 80 (2002), no. 1, 73-100.
• [25] E.A. Nadaraya, On estimating regression, Theor. Probab. Appl. 9 (1964), 141–142.
• [26] H. Park, L.A. Stefanski, Relative error prediction, Statist. Probab. Lett. 40 (1998), no. 3, 227–236.
• [27] M. Pawlak, E. Rafajłowicz, Jump preserving signal reconstruction using vertical weighting, Nonlinear Anal. 47 (2001), no. 1, 327-338.
• [28] E. Rafajłowicz, M. Pawlak, A. Steland, Nonlinear image processing and filtering: a unified approach based on vertically weighted regression, Int. J. Appl. Math. Comput. Sci. 18 (2008), no. 1, 49–61.
• [29] M. Rosenblatt, Remark on some nonparametric estimates of density function, Ann. Math. Statist. 27 (1956), 832–837.
• [30] B. Thiam, Relative error prediction in nonparametric deconvolution regression model, Stat. Neerl. 73 (2019), no. 1, 63-77.
• [31] G.S. Watson, Smooth regression analysis, Sankhya Ser. A 26 (1964), 359–372.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).