Local linear estimation of the conditional cumulative distribution function: censored functional data case. (English) Zbl 07730318
Sankhyā, Ser. A 85, No. 1, 741-769 (2023); correction ibid. 85, No. 1, 770 (2023).
Summary: In this paper, we estimate the conditional cumulative distribution function of a randomly censored scalar response variable given a functional random variable using the local linear approach. Under this structure, we state the asymptotic normality with explicit rates of the constructed estimator. Moreover, the usefulness of our results is illustrated through a simulated study.
MSC:
| 62G05 | Nonparametric estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G30 | Order statistics; empirical distribution functions |
| 62N01 | Censored data models |
| 62N02 | Estimation in survival analysis and censored data |
Keywords:
asymptotic normality; censored data; conditional cumulative distribution; functional data; local linear estimationReferences:
| [1] | Altendji, B.; Demongeot, J.; Laksaci, A.; Rachdi, M., Functional data analysis: estimation of the relative error in functional regression under random left-truncation model, Journal of Nonparametric Statistics, 30, 472-490 (2018) · Zbl 1408.62064 · doi:10.1080/10485252.2018.1438609 |
| [2] | Ayad, S.; Laksaci, A.; Rahmani, S.; Rouane, R., On the local linear modelization of the conditional density for functional and ergodic data, METRON, 78, 237-254 (2020) · Zbl 1452.62262 · doi:10.1007/s40300-020-00174-6 |
| [3] | Al-Awadhi, F.; Kaid, A.; Laksaci, Z.; Ouassou, I.; Rachdi, M., Functional data analysis: local linear estimation of the L1,-conditional quantiles, Statistical Methods and Applications, 28, 217-240 (2019) · Zbl 1427.62146 · doi:10.1007/s10260-018-00447-5 |
| [4] | Baìllo, A., Local linear regression for functional predictor and scalar response, J. of Multivariate Anal, 100, 102-111 (2009) · Zbl 1151.62028 · doi:10.1016/j.jmva.2008.03.008 |
| [5] | Barrientos, J.; Ferraty, F.; Vieu, P., Locally Modelled Regression Functional Data, J. Nonparametr. Statist, 22, 617-632 (2010) · Zbl 1327.62191 · doi:10.1080/10485250903089930 |
| [6] | Beran, R (1981). Nonparametric Regression with Randomly Censored Survival Data, Technical report, University of California, Berkeley. |
| [7] | Berlinet, A.; Elamine, A.; Mas, A., Local linear regression for functional data, Inst. Statist. Math., 63, 1047-1075 (2011) · Zbl 1225.62093 · doi:10.1007/s10463-010-0275-8 |
| [8] | Bouanani, O.; Laksaci, A.; Rachdi, M.; Rahmani, S., Asymptotic normality of some conditional nonparametric functional parameters in high-dimensional statistics, Behaviormetrika, 46, 199-233 (2019) · doi:10.1007/s41237-018-0057-9 |
| [9] | Demongeot, J.; Laksaci, A.; Madani, F.; Rachdi, M., Functional data: local linear estimation of the conditional density and its application, Statistics, 47, 26-44 (2013) · Zbl 1440.62117 · doi:10.1080/02331888.2011.568117 |
| [10] | Demongeot, J.; Laksaci, A.; Rachdi, M.; Rahmani, S., On the local linear modelization of the conditional distribution for functional data, Sankhya, 76, 328-355 (2014) · Zbl 1307.62093 · doi:10.1007/s13171-013-0050-z |
| [11] | Demongeot, J.; Laksaci, A.; Naceri, A.; Rachdi, M., Local Linear RegressionModelization When All Variables are Curves, Statistics and Probability Letters, 121, 37-44 (2017) · Zbl 1351.62083 · doi:10.1016/j.spl.2016.09.021 |
| [12] | Fan, J.; Gijbels, I., Censored regression: local linear approximations and their applications, Journal of the American Statistical Association, 89, 560-570 (1994) · Zbl 0802.62044 · doi:10.1080/01621459.1994.10476781 |
| [13] | Ferraty, F.; Laksaci, A.; Vieu, P., Estimation of some characteristics of the conditional distribution in nonparametric functional models, Statist. Inference Stoch. Process., 9, 47-76 (2006) · Zbl 1117.62030 · doi:10.1007/s11203-004-3561-3 |
| [14] | Ferraty, F.; Vieu, P., Nonparametric functional data analysis: Theory and Practice (2006), New York: Springer Series in Statistics, New York · Zbl 1119.62046 |
| [15] | Ferraty, F.; Mas, A.; Vieu, P., Nonparametric regression on functional data: inference and practical aspects, Australian New Zealand J. Statistics, 49, 267-286 (2007) · Zbl 1136.62031 · doi:10.1111/j.1467-842X.2007.00480.x |
| [16] | Helal, N.; Ould-Saïd, E., Kernel conditional quantile estimator under left truncation for functional regressors, Opuscula Mathematica, 36, 25-48 (2016) · Zbl 1329.62167 · doi:10.7494/OpMath.2016.36.1.25 |
| [17] | Horrigue, W.; Ould-Saïd, E., Strong uniform consistency of a nonparametric estimator of a conditional quantile for censored dependent data and functional regressors, Random Operators and Stochastic Equations, 19, 131-156 (2011) · Zbl 1348.62169 · doi:10.1515/ROSE.2011.008 |
| [18] | Horrigue, W.; Ould-Saïd, E., Nonparametric regression quantile estimation for dependant functional data under random censorship: Asymptotic normality, Communications in Statistics - Theory and Methods, 44, 4307-4332 (2014) · Zbl 1333.62101 · doi:10.1080/03610926.2013.784993 |
| [19] | Kaplan, EM; Meier, P., Nonparametric estimation from incomplete observations, Journal of the American Statistical Association, 53, 457-481 (1958) · Zbl 0089.14801 · doi:10.1080/01621459.1958.10501452 |
| [20] | Laksaci, A.; Rachdi, M.; Rahmani, S., Spatial modelization: local linear estimation of the conditional distribution for functional data, Spat. Statist., 6, 1-23 (2013) · doi:10.1016/j.spasta.2013.04.004 |
| [21] | Leulmi, S (2019). Local linear estimation of the conditional quantile for censored data and functional regressors. Communications in Statistics-Theory and Methods, 1-15. |
| [22] | Lipsitz, SR; Ibrahim, JG, Estimation with Correlated Censored Survival Data with Missing Covariates, Biostatistics, 1, 315-27 (2000) · Zbl 0965.62080 · doi:10.1093/biostatistics/1.3.315 |
| [23] | Ling, N.; Liu, Y.; Vieu, P., Nonparametric regression estimation for functional stationary ergodic data with missing at random, Journal of Statistical Planning and Inference, 162, 75-87 (2015) · Zbl 1314.62102 · doi:10.1016/j.jspi.2015.02.001 |
| [24] | Ling, N.; Liu, Y.; Vieu, P., Conditional mode estimation for functional stationary ergodic data with responses missing at random, Statistics, 50, 991-1013 (2016) · Zbl 1358.62036 · doi:10.1080/02331888.2015.1122012 |
| [25] | Li, WV; Shao, QM, Gaussian processes: inequalities, small ball probabilities and applications, Handbook of Statistics, 19, 533-597 (2001) · Zbl 0987.60053 · doi:10.1016/S0169-7161(01)19019-X |
| [26] | Elias Ould, S.; Sadki, Q., Asymptotic normality for a smooth kernel estimator of the conditional quantile for censored time series, South African Statistical Journal, 45, 65-98 (2011) · Zbl 1397.62165 |
| [27] | Kohler, M.; Kinga, M.; Márta, P., Prediction from randomly right censored data, Journal of Multivariate Analysis, 80, 73-100 (2002) · Zbl 0992.62041 · doi:10.1006/jmva.2000.1973 |
| [28] | Kohler, M.; Màthè, K.; Pintèr, M., Prediction from randomly right censored data, Journal of Multivariate Analysis, 80, 73-100 (2002) · Zbl 0992.62041 · doi:10.1006/jmva.2000.1973 |
| [29] | Ren, JJ; Gu, M., Regression M-estimators with doubly censored data, Ann. Statist, 25, 2638-2664 (1997) · Zbl 0907.62045 · doi:10.1214/aos/1030741089 |
| [30] | Stute, W., Consistent estimation under random censorship when covariables are present, J. Multivariate Anal., 45, 89-103 (1993) · Zbl 0767.62036 · doi:10.1006/jmva.1993.1028 |
| [31] | Zhou, ZY; Lin, Z., Asymptotic normality of locally modelled regression estimator for functional data, Nonparametric Stat., 28, 116-131 (2016) · Zbl 1381.62064 · doi:10.1080/10485252.2015.1114112 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
