A Bernstein polynomial approach to the estimation of a distribution function and quantiles under censorship model

References

• Babu, G. J., A. J. Canty, and Y. P. Chaubey. 2002. Application of Bernstein polynomials for smooth estimation of a distribution and density function. Journal of Statistical Planning and Inference 105 (2):377–92. doi: 10.1016/S0378-3758(01)00265-8.
   Web of Science ®Google Scholar
• Babu, G. J., and Y. P. Chaubey. 2006. Smooth estimation of a distribution and density function on a hypercube using Bernstein polynomials for dependent random vectors. Statistics & Probability Letters 76 (9):959–69. doi: 10.1016/j.spl.2005.10.031.
   Web of Science ®Google Scholar
• Beran, R. 1981. Nonparametric regression with randomly censored survival data. Technical report, University of California, Berkeley.
   Google Scholar
• Bernstein, S. 1912. Démonstration du théorème de weier-strass fondée sur le calcul des probabilités. Community of the Kharkov Mathematical Society 13:1–2.
   Google Scholar
• Bouaziz, O., and O. Lopez. 2010. Conditional density estimation in a censored single-index regression model. Bernoulli 16 (2):514–42. doi: 10.3150/09-BEJ221.
   Web of Science ®Google Scholar
• Brunel, E., F. Comte, and C. Lacour. 2007. Adaptive estimation of the conditional density in the presence of censoring. Sankhya 69 (4):734–63.
   Google Scholar
• Bouezmarni, T., A. El Ghouch, and M. Mesfioui. 2011. Gamma kernel estimators for density and hazard rate of right-censored data. Journal of Probability and Statistics 2011:1–16. doi: 10.1155/2011/937574.
   Google Scholar
• Chaouch, M., and S. Khardani. 2015. Randomly censored quantile regression estimation using functional stationary ergodic data. Journal of Nonparametric Statistics 27 (1):65–87. doi: 10.1080/10485252.2014.982651.
   Web of Science ®Google Scholar
• Dabrowska, D. M. 1987. Nonparametric regression with censored survival time data. Scandinavian Journal of Statistics 14:181–97.
   Web of Science ®Google Scholar
• Dabrowska, D. M. 1989. Uniform consistency of the kernel conditional Kaplan Meier estimate. Annals of Statistics 17:1157–67.
   Web of Science ®Google Scholar
• Deheuvels, P., and J. H. Einmahl. 2000. Functional limit laws of Kaplan Meier product-limit processes and applications. The Annals of Probability 28:1301–35. doi: 10.1214/aop/1019160336.
   Web of Science ®Google Scholar
• Dib, K., T. Bouezmarni, M. Belalia, and A. Kitouni. 2020. Nonparametric bivariate distribution estimation using Bernstein polynomials under right censoring. Communications in Statistics-Theory and Methods 49:1–11.
   Google Scholar
• Helali, S., Y. Slaoui, and A. Masmoudi, 2022. Semi parametric estimation using Bernstein polynomial and a finite Gaussian mixture model. Entropy 24 (3):315. doi: 10.3390/e24030315.
   PubMed Web of Science ®Google Scholar
• Jmaei, A., Y. Slaoui, and W. Dellagi. 2017. Recursive distribution estimator defined by stochastic approximation method using Bernstein polynomials. Journal of Nonparametric Statistics 29 (4):792–805. doi: 10.1080/10485252.2017.1369538.
   Web of Science ®Google Scholar
• Khardani, S., M. Lemdani, and E. Ould Saïd. 2011. Uniform rate of strong consistency for a smooth kernel estimator of the conditional mode for censored time series. Journal of Statistical Planning and Inference 141 (11):3426–36. doi: 10.1016/j.jspi.2011.04.023.
   Web of Science ®Google Scholar
• Khardani, S., and S. Semmar. 2014. Nonparametric conditionel density estimation for censored data based on a recursive kernel. The Electronic Journal of Statistics 8:2541–56. doi: 10.1214/14-EJS960.
   Web of Science ®Google Scholar
• Khardani, S., M. Lemdani, and E. Ould Saïd. 2010. Some asymptotic properties for a smooth kernel estimator of the conditional mode under random censorship. Journal of the Korean Statistical Society 39 (4):455–69. doi: 10.1016/j.jkss.2009.10.001.
   Web of Science ®Google Scholar
• Khardani, S., M. Lemdani, and E. Ould Saïd. 2012. On the strong uniform consistency of the mode estimator for censored time series. Metrika 75 (2):229–41. doi: 10.1007/s00184-010-0324-6.
   Web of Science ®Google Scholar
• Lacksaci, A., S. Khardani, and S. Semmar. 2020. Semi-recursive kernel conditional density estimators under alpha mixing data. Communications in Statistics - Theory and Methods 51 (7):2116–38. doi: 10.1080/03610926.2020.1764038.
   Web of Science ®Google Scholar
• Leblanc, A. 2010. A bias-reduced approach to density estimation using Bernstein polynomials. Journal of Nonparametric Statistics 22 (4):459–75. doi: 10.1080/10485250903318107.
   Web of Science ®Google Scholar
• Leblanc, A. 2012. On estimating distribution functions using Bernstein polynomials. Annals of the Institute of Statistical Mathematics 64 (5):919–43. doi: 10.1007/s10463-011-0339-4.
   Web of Science ®Google Scholar
• Lopez, O., and P. Saint-Pierre. 2012. Bivariate censored regression relying on a new estimator of the joint distribution function. Journal of Statistical Planning and Inference 142 (8):2440–53. doi: 10.1016/j.jspi.2012.02.046.
   Web of Science ®Google Scholar
• Slaoui, Y. 2014a. Bandwidth selection for recursive kernel density estimators defined by stochastic approximation method. Journal of Probability and Statistics 2014:1–11. doi: 10.1155/2014/739640.
   Google Scholar
• Slaoui, Y. 2014b. The stochastic approximation method for estimation of a distribution function. Mathematical Methods of Statistics 23 (4):306–25. doi: 10.3103/S1066530714040048.
   Google Scholar
• Slaoui, Y, and S. Khardani. 2020. Adaptive recursive kernel conditional density estimators under censoring data. Latin American Journal of Probability and Mathematical Statistics 17 (1):389 doi: 10.30757/ALEA.v17-16.
   Google Scholar
• Slaoui, Y. 2022. Moderate deviation principles for nonparametric recursive distribution estimators using Bernstein polynomials. Revista Matematica Complutense 35:145–58.
   Web of Science ®Google Scholar
• Slaoui, Y. 2022. Bernstein polynomial of recursive regression estimation with censored data. Stochastic Models 38:468–87. doi: 10.1080/15326349.2022.2063335.
   Web of Science ®Google Scholar
• Soni, P., I. Dewan, and K. Jain. 2012. Nonparametric estimation of quantile density function. Computational Statistics & Data Analysis 56 (12):3876–86. doi: 10.1016/j.csda.2012.04.014.
   Web of Science ®Google Scholar
• Stute, W. 1993. Consistent estimation under random censorship when covariables are present. Journal of Multivariate Analysis 45 (1):89–103. doi: 10.1006/jmva.1993.1028.
   Web of Science ®Google Scholar
• Vitale, R. 1975. A Bernstein polynomial approach to density estimation. Statistical Inference and Related Topics, vol. 2, 87– 99.
   Google Scholar