## Bernoulli

• Bernoulli
• Volume 19, Number 3 (2013), 721-747.

### Single index regression models in the presence of censoring depending on the covariates

#### Abstract

Consider a random vector $(X',Y)'$, where $X$ is $d$-dimensional and $Y$ is one-dimensional. We assume that $Y$ is subject to random right censoring. The aim of this paper is twofold. First, we propose a new estimator of the joint distribution of $(X',Y)'$. This estimator overcomes the common curse-of-dimensionality problem, by using a new dimension reduction technique. Second, we assume that the relation between $X$ and $Y$ is given by a mean regression single index model, and propose a new estimator of the parameters in this model. The asymptotic properties of all proposed estimators are obtained.

#### Article information

Source
Bernoulli, Volume 19, Number 3 (2013), 721-747.

Dates
First available in Project Euclid: 26 June 2013

https://projecteuclid.org/euclid.bj/1372251141

Digital Object Identifier
doi:10.3150/12-BEJ464

Mathematical Reviews number (MathSciNet)
MR3079294

Zentralblatt MATH identifier
1273.62089

#### Citation

Lopez, Olivier; Patilea, Valentin; Van Keilegom, Ingrid. Single index regression models in the presence of censoring depending on the covariates. Bernoulli 19 (2013), no. 3, 721--747. doi:10.3150/12-BEJ464. https://projecteuclid.org/euclid.bj/1372251141

#### References

• [1] Akritas, M.G. (1994). Nearest neighbor estimation of a bivariate distribution under random censoring. Ann. Statist. 22 1299–1327.
• [2] Akritas, M.G. and Van Keilegom, I. (2000). The least squares method in heteroscedastic censored regression models. In Asymptotics in Statistics and Probability (M.L. Puri, ed.) 379–391. Utrecht: VSP.
• [3] Andersen, P.K. and Gill, R.D. (1982). Cox’s regression model for counting processes: A large sample study. Ann. Statist. 10 1100–1120.
• [4] Beran, R. (1981). Nonparametric regression with randomly censored survival data. Technical report, Univ. California, Berkeley.
• [5] Delecroix, M., Hristache, M. and Patilea, V. (2006). On semiparametric $M$-estimation in single-index regression. J. Statist. Plann. Inference 136 730–769.
• [6] Dominitz, J. and Sherman, R.P. (2005). Some convergence theory for iterative estimation procedures with an application to semiparametric estimation. Econometric Theory 21 838–863.
• [7] Du, Y. and Akritas, M.G. (2002). Uniform strong representation of the conditional Kaplan–Meier process. Math. Methods Statist. 11 152–182.
• [8] Einmahl, U. and Mason, D.M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33 1380–1403.
• [9] Fan, J. and Gijbels, I. (1994). Censored regression: Local linear approximations and their applications. J. Amer. Statist. Assoc. 89 560–570.
• [10] Gørgens, T. and Horowitz, J.L. (1999). Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable. J. Econometrics 90 155–191.
• [11] Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models. Ann. Statist. 21 157–178.
• [12] Härdle, W. and Stoker, T.M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84 986–995.
• [13] Heuchenne, C. and Van Keilegom, I. (2007). Polynomial regression with censored data based on preliminary nonparametric estimation. Ann. Inst. Statist. Math. 59 273–297.
• [14] Horowitz, J.L. and Härdle, W. (1996). Direct semiparametric estimation of single-index models with discrete covariates. J. Amer. Statist. Assoc. 91 1632–1640.
• [15] Hristache, M., Juditsky, A. and Spokoiny, V. (2001). Direct estimation of the index coefficient in a single-index model. Ann. Statist. 29 595–623.
• [16] Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120.
• [17] Klein, R.W. and Spady, R.H. (1993). An efficient semiparametric estimator for binary response models. Econometrica 61 387–421.
• [18] Li, K.C., Wang, J.L. and Chen, C.H. (1999). Dimension reduction for censored regression data. Ann. Statist. 27 1–23.
• [19] Lopez, O. (2007). Réduction de dimension en présence de données censurées. Ph.D. thesis, CREST-ENSAI. Available at http://tel.archives-ouvertes.fr/tel-00195261/fr.
• [20] Lopez, O. (2009). Single-index regression models with right-censored responses. J. Statist. Plann. Inference 139 1082–1097.
• [21] Lopez, O. (2011). Nonparametric estimation of the multivariate distribution function in a censored regression model with applications. Comm. Statist. Theory Methods 40 2639–2660.
• [22] Lu, X. and Burke, M.D. (2005). Censored multiple regression by the method of average derivatives. J. Multivariate Anal. 95 182–205.
• [23] Lu, X. and Cheng, T.L. (2007). Randomly censored partially linear single-index models. J. Multivariate Anal. 98 1895–1922.
• [24] Nolan, D. and Pollard, D. (1987). $U$-processes: Rates of convergence. Ann. Statist. 15 780–799.
• [25] Powell, J.L., Stock, J.H. and Stoker, T.M. (1989). Semiparametric estimation of index coefficients. Econometrica 57 1403–1430.
• [26] Sánchez Sellero, C., González Manteiga, W. and Van Keilegom, I. (2005). Uniform representation of product-limit integrals with applications. Scand. J. Statist. 32 563–581.
• [27] Sherman, R.P. (1994). Maximal inequalities for degenerate $U$-processes with applications to optimization estimators. Ann. Statist. 22 439–459.
• [28] Stute, W. (1993). Consistent estimation under random censorship when covariables are present. J. Multivariate Anal. 45 89–103.
• [29] Stute, W. (1996). Distributional convergence under random censorship when covariables are present. Scand. J. Statist. 23 461–471.
• [30] Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22 28–76.
• [31] van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics 3. Cambridge: Cambridge Univ. Press.
• [32] van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer Series in Statistics. New York: Springer.
• [33] Van Keilegom, I. and Akritas, M.G. (1999). Transfer of tail information in censored regression models. Ann. Statist. 27 1745–1784.
• [34] Wang, Y., He, S., Zhu, L. and Yuen, K.C. (2007). Asymptotics for a censored generalized linear model with unknown link function. Probab. Theory Related Fields 138 235–267.