Efficient Estimation Of Left-Truncated And Right Censored Data With Additive Hazard Model


  • Wissam Kamil Ghafil Kamil Ghafil Department of Mathematics, University of the-Qar, Iraq
  • Riyadh R. Al-Mosawi Department of Mathematics, Thi-Qar University, Iraq.




In this paper, an additive hazard model is considered as a semiparametric model in which the unknown
parameter consists of finite dimensional and infinite dimensional parts with left truncated and right
censored data. The full likelihood function for the model is obtained for the parametric part and also for
the nonparametric part using linear sieve procedure and then compute the maximum likelihood estimators
for the two parts. The consistency of the maximum likelihood estimators is also proved for the two type
of parameters. The score operators for the parametric and nonparametric parts are obtained and their
adjoint score operators are computed. Finally, a simulation study using Monti-Carlo method and R
language is implemented to compute the maximum likelihood estimators and compare the results of the
proposed method with the true values. As a real life application, Stanford Heart transplant data is
considered and the maximum likelihood estimators are computed


Aalen, O. O. (1989). A Linear Regression Model for the Analysis of Life Times. Statisticin

Medicine, 8, 907-925.

Chen, L. and Sun, J. (2009). A multiple imputation approach to the analysis of current

status data with the additive hazards model. Communications in Statistics-Theory and

Methods, 38, 1009-1018.

Feng, Y., Ma, L . and Sun, J. (2014). Regression analysis of current status data under

the additive hazard model with auxiliary covariates. Jornal of Statistics, 42, 118-136.

Wang, L. Sun, J. and Tong, X. (2010). Regression analysis of case II interval-censored

failure time data with the additive hazards model. Statistical Sinica, 20, 1709-1723.

Lu, X. and Peter, X.K. (2014). Efficient estimation of the partly linear additive hazards

model with current status data. Jornal of Statistics, doi:10.1111/sjos.12108.

Shen, S.P. (2014). A Generalization of turnbull’s estimator for interval-censored and

truncated data. Communications in Statistics, 43, 2958-2972.

Huang, J. (1999). Efficient estimation of the partly linear additive cox model. The

annals of Statistics, 27(5), 1536-1563.

Pan, W. and Chappell, R. (2002). Estimation in the Cox proportional hazards model

with left truncated and interval censored data. Jornal of the Inter, 58(1), 64-70.

Kim, J.S. (2003). Efficient estimation for the proportional hazards model with lefttruncated and case

interval-censored data. Statistica Sinica, 13, 519-537.

Zeng, D. Cai, J. and Shen, Y. (2006). Semiparametric additive risks model for intervalcensored

data. Statistical Sinica, 16, 287-302.

Wang, L. Sun, J. and Tong, X. (2010). Regression analysis of case II interval-censored

failure time data with the additive hazards model. Statistica Sinica, Vol, 20, 1709-1723.

Su, Y.R. and Wang, J.L. (2012). Modeling left-truncated and right-censored survival

data with longitudinal covariates. The Annals of Statistics, 40(3), 1465-1488.14

Lin, D. Y. Oakesm D. and Ying, Z. (1998). Additive hazarde regression with current

status data. Biometrika, 85(2), 289-298.

Huang, J and Wellner, J. A. (1996). Interval censored survival data. Biostatistics, 123, 123-169.

Hang, J. (1995). Maximum likelihood esttimation for Proportional Odds Regression

Model with Current Status Data. Monograph series, 27.

Lin, D. Y. and Ying, Z. (1994). Semiparametric analysis of the additive risk model.

Biometrika, 81(1), 61-71.

Song, X., Sun, L., Mu, X. and Dinse, E. G. (2011). Additive hazareds regression with

censoring indictors missing at random. Can. J. Stat.,38(3), 333-351.

Qin, J. (2013). Semiparametric esttimation for the additive hazareds model with lefttruncated and

right censored data. Biometrika, 100(4), 877-888.

Shen, P. S. (2015). Semiparametric Analysis of Transformation Models with Dependently Left truncated

and Right-censored Data. Communications in StatisticsComputation and Simulation,


Shen, P. S. (2014). Proportional hazards regression with interval censored lefttruncated data.

Journal of Statistical Computation and Simulation.

Shen, P. S. (2014). Aalens Additive Risk Model for Left-Truncated and Right-Censored

Data, Journal of Statistical Computation and Simulation.

Chen, L. P. (2016). Pseudo Likelihood Estimation for the Additive Hazards Model

with Data subject to LeftTruncation and Right-Censoring.

Ma, L., Hu, T. and Sun, J. (2015). Sieve maximum likelihood regression analysis of

dependent current status data. Biometrika, 18 doi: 10.1093/biomet/asv020.

Feng, Y. and Sun, J. (2014). Regession Analysis of current status data under thr

additive hazards model with auxiliary covariates. Scandinavian Journal of Statistics,

doi: 10.1111/sjos.12098.15