Estimation of Cumulative Incidence Function in the Presence of Middle Censoring Using Improper Gompertz Distribution




Cumulative incidence function, Improper Gompertz distribution, Middle censoring, Maximum likelihood estimator, Bayes estimators, MCMC method


In this paper we deal with the modelling of cumulative incidence function using improper Gompertz distribution based on middle censored competing risks survival data. Together with the unknown parameters, cumulative incidence function also estimated. In classical set up, we derive the point estimates using maximum likelihood estimator and midpoint approximation methods. The asymptotic confidence interval are obtained based on asymptotic normality properties of maximum likelihood estimator. We also derive the Bayes estimates with associated credible intervals based on informative and non-informative types of priors under two loss functions such as squared error and LINEX loss functions. A simulation study is conducted for comprehensive comparison between various estimators proposed in this paper. A real life data set is also used for illustration.


A. ABUZAID, M. A. EL-QUMSAN, A. EL-HABIL (2017). On the robustness of right and middle censoring schemes in parametric survival models. Communications in Statistics-Simulation and Computation, 46, no. 3, pp. 1771–1780.

K. AHMADI, M. REZAEI, F. YOUSEFZADEH (2017). Statistical analysis of middle censored competing risks data with exponential distribution. Journal of Statistical Computation and Simulation, 87, no. 16, pp. 3082–3110.

D. CHEN, Y. LIO (2010). Parameter estimations for generalized exponential distribution under progressive type-I interval censoring. Computational Statistics & Data Analysis, 54, no. 6, pp. 1581–1591.

J. P. FINE, R. J. GRAY (1999). A proportional hazards model for the subdistribution of a competing risk. Journal of the American statistical association, 94, no. 446, pp. 496– 509.

S.GEMAN, D.GEMAN (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, no. 6, pp. 721–741.

B. GOMPERTZ (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, , no. 115, pp. 513-583.

S. HAILE, J.-H. JEONG, X. CHEN, Y. CHENG (2016). A 3-parameter Gompertz distribution for survival data with competing risks, with an application to breast cancer data. Journal of Applied Statistics, 43, no. 12, pp. 2239–2253.

W. K. HASTINGS (1970). Monte carlo sampling methods using markov chains and their applications. Biometrika, 57, no. 1, pp. 97–109.

S. R. JAMMALAMADAKA, V.MANGALAM (2003). Nonparametric estimation for middlecensored data. Journal of Nonparametric Statistics, 15, no. 2, pp. 253–265.

J.-H. JEONG, J. FINE (2006). Direct parametric inference for the cumulative incidence function. Journal of the Royal Statistical Society: Series C (Applied Statistics), 55, no. 2, pp. 187–200.

J.-H. JEONG, J. P. FINE (2007). Parametric regression on cumulative incidence function. Biostatistics, 8, no. 2, pp. 184–196.

J. D. KALBFLEISCH, R. L. PRENTICE (2002). The Statistical Analysis of Failure Time Data, vol. 360. JohnWiley & Sons, New Jersey.

W. R. KIM, T. M. THERNEAU, J. T. BENSON, W. K. KREMERS, C. B. ROSEN, G. J. GORES, E. R. DICKSON (2006). Deaths on the liver transplant waiting list: an analysis of competing risks. Hepatology, 43, no. 2, pp. 345–351.

M. LEE (2019). Parametric inference for quantile event times with adjustment for covariates on competing risks data. Journal of Applied Statistics, 46, no. 12, pp. 2128–2144.

D. LUNN, C. JACKSON,N. BEST, D. SPIEGELHALTER, A. THOMAS (2012). The BUGS Book: A Practical Introduction to Bayesian Analysis. Chapman and Hall/CRC, Boca Raton.

A. W. MARSHALL, I. OLKIN (2007). Life distributions, vol. 13. Springer-Verlag, New York.

C. P. ROBERT, G. CASELLA (2010). Introducing Monte Carlo Methods with R, vol. 18. Springer-Verlag, New York.

M. TEIMOURI, A. K. GUPTA (2012). Estimation methods for the Gompertz–Makeham distribution under progressively type-I interval censoring scheme. National Academy Science Letters, 35, no. 3, pp. 227–235.

A. TSIATIS (1975). A nonidentifiability aspect of the problem of competing risks. Proceedings of the National Academy of Sciences, 72, no. 1, pp. 20–22.

L.WANG (2016). Estimation for exponential distribution based on competing risk middle censored data. Communications in Statistics-Theory and Methods, 45, no. 8, pp. 2378–2391.

W. YAN, S. YIMIN, W. MIN (2019). Statistical inference for dependence competing risks model under middle censoring. Journal of Systems Engineering and Electronics, 30, no. 1, pp. 209–222.




How to Cite

Rehman, H., & Chandra, N. (2021). Estimation of Cumulative Incidence Function in the Presence of Middle Censoring Using Improper Gompertz Distribution. Statistica, 81(2), 163–182.