Abstract
This paper considers the estimation of a semiparametric isotonic regression model when the covariates are measured with additive errors and the response is randomly right censored by a censoring time. The authors show that the proposed estimator of the regression parameter is rootn consistent and asymptotically normal. The authors also show that the isotonic estimator of the functional component, at a fixed point, is cubic root-n consistent and converges in distribution to the slope at zero of the greatest convex minorant of the sum of a two-sided standard Brownian motion and the square of the time parameter. A simulation study is carried out to investigate the performance of the estimators proposed in this article.
Similar content being viewed by others
References
Huang J, A note on estimating a partly linear model under monotonicity constraints, Journal of Statistical Planning and Inference, 2002, 107(1–2): 345–351.
Cheng G, Semiparametric additive isotonic regression, Journal of Statistical Planning and Inference, 2009, 139(6): 1980–1991.
Brunk H D, On the estimation of parameters restricted by inequalities, Ann. Math. Statist., 1958, 29(2): 437–454.
Barlow R E, Bartholomew D J, Bremner J M, and Brunk H D, Statistical Inference Under Order Restrictions, Wiley, New York, 1972.
Wright F T, The asymptotic behavior of monotone regression estimates, The Annals of Statistics, 1981, 9(2): 443–448.
Robertson T, Wright F T, and Dykstra R, Order Restricted Statistical Inference, John Wiley and Sons, New York, 1988.
Hall P and Huang L S, Nonparametric kernel regression subject to monotonicity constraints, The Annals of Statistics, 2001, 29(3): 624–647.
Mammen E, Marron J S, Turlach B A, and Wand M P, A general projection framework for constrained smoothing, Statist. Sci., 2001, 16(3): 232–248.
Dette H, Neumeyer N, and Pilz K F, A simple nonparametric estimator of a strictly monotone regression function, Bernoulli, 2006, 12(3): 469–490.
Miller R G, Least squares regression with censored data, Biometrika, 1976, 63(3): 449–464.
Buckley J and James J, Linear regression with censored data, Biometrika, 1979, 66(3): 429–436.
Koul H, Susarla V, and Ryzin J V, Regression analysis with randomly right-censored data, The Annals of Statistics, 1981, 9(6): 1276–1288.
Leurgans S, Linear modes, random censoring, and synthetic data, Biometrika, 1987, 74(2): 301–309.
Zhou M, Asymptotic normality of the ‘synthetic data’ regression estimator for censored survival data, The Annals of Statistics, 1992, 20(2): 1002–1021.
Lai T L, Ying Z L, and Zheng Z K, Asymptotic normality of a class of adaptive statistics with applications to synthetic data method for censored regression, Journal of Multivariate Analysis, 1995, 52(2): 259–279.
Zhao P X and Xue L G, Variable selection for semiparametric varying coefficient partially linear errors-in-variables models, Journal of Multivariate Analysis, 2010, 101(8): 1872–1883.
Liu Q and Xue L G, Empirical likelihood-based inference for single-index EV models with validation data, Chinese Journal of Engineering Mathematics, 2010, 27(2): 321–332.
Liang H, Wang S, and Carroll R J, Partially linear models with missing response variables and error-prone covariates, Biometrika, 2007, 94(1): 185–198.
Liang H, Härdle W, and Carroll R J, Estimation in a semiparametric partially linear errors-invariables model, The Annals of Statistics, 1999, 27(5): 1519–1535.
Carroll R J, Ruppert D, and Stefanski L A, Measurement Error in Nonlinear Models, Chapman and Hall, New York, 1995.
Shi J and Lau T S, Empirical likelihood for partially linear models, Journal of Multivariate Analysis, 2000, 72(1): 132–148.
Cui H J and Kong E F, Empirical likelihood confidence region for parameters in semi-linear errors-in-variables models, Scand. J. Statist., 2006, 33(1): 153–168.
Shorack G R and Wellner J A, Empirical Process with Applications to Statistics, John Wiley and Sons, New York, 1986.
Lin Z Y, Lu C R, and Su Z G, Introduction of Probability Limit Theory, Higher Education Press, Beijing, 1999.
Van der Vaart A W and Wellner J A, Weak Convergence and Empirical Processes: With Application to Statistics, Springer, New York, 1996.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China under Grant No. 10971007; Foundation of Academic Discipline Program at Central University of Finance and Economics; Funding Project of Science and Technology Research Plan of Beijing Education Committee under Grant No. 00600054K1002; Fund of 211 Project at Central University of Finance and Economics; 2012 National Project of Statistical Research.
This paper was recommended for publication by Editor ZOU Guohua.
Rights and permissions
About this article
Cite this article
Sun, Z., Zhang, Z. & Du, J. Semiparametric analysis of isotonic errors-in-variables regression models with randomly right censored response. J Syst Sci Complex 26, 441–461 (2013). https://doi.org/10.1007/s11424-013-1154-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-013-1154-1