Lifetime Data Analysis

, Volume 17, Issue 4, pp 594–607 | Cite as

Nonparametric quasi-likelihood for right censored data



Quasi-likelihood was extended to right censored data to handle heteroscedasticity in the frame of the accelerated failure time (AFT) model. However, the assumption of known variance function in the quasi-likelihood for right censored data is usually unrealistic. In this paper, we propose a nonparametric quasi-likelihood by replacing the specified variance function with a nonparametric variance function estimator. This nonparametric variance function estimator is obtained by smoothing a function of squared residuals via local polynomial regression. The rate of convergence of the nonparametric variance function estimator and the asymptotic limiting distributions of the regression coefficient estimators are derived. It is demonstrated in simulations that for finite samples the proposed nonparametric quasi-likelihood method performs well. The new method is illustrated with one real dataset.


Kaplan–Meier estimate Local polynomial smoothing Semiparametric modeling Survival analysis Variance function 


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  1. Buckley J, James I (1979) Linear-regression with censored data. Biometrika 66(3): 429–436CrossRefMATHGoogle Scholar
  2. Chiou JM, Muller HG (1999) Nonparametric quasi-likelihood. Ann Stat 27(1): 36–64CrossRefMATHMathSciNetGoogle Scholar
  3. Cox DR (1972) Regression models and life-tables. J R Stat Soc B Stat Methodol 34(2): 187–220MATHGoogle Scholar
  4. Cox DR, Oakes D (1984) Analysis of survival data. Chapman and Hall, LondonGoogle Scholar
  5. Fahrmeir L (1990) Maximum likelihood estimation in misspecified generalized linear models. Statistics 21: 487–502CrossRefMATHMathSciNetGoogle Scholar
  6. Fan JQ (1992) Design-adaptive nonparametric regression. J Am Stat Assoc 87(420): 998–1004CrossRefMATHGoogle Scholar
  7. Fan JQ, Gijbels I (1992) Variable bandwidth and local linear-regression smoothers. Ann Stat 20(4): 2008–2036CrossRefMATHMathSciNetGoogle Scholar
  8. Fan J, Gijbels I (1996) Local polynomial modelling and its applications. Chapman and Hall/CRC, LondonMATHGoogle Scholar
  9. James IR (1986) On estimating equations with censored-data. Biometrika 73(1): 35–42CrossRefMATHMathSciNetGoogle Scholar
  10. Jin ZZ et al (2003) Rank-based inference for the accelerated failure time model. Biometrika 90(2): 341–353CrossRefMATHMathSciNetGoogle Scholar
  11. Kalbfleisch DJ, Prentice LR (1980) The statistical analysis of failure time data. Wiley, New YorkMATHGoogle Scholar
  12. Krall JM, Uthoff VA, Harley JB (1975) Step-up procedure for selecting variables associated with survival. Biometrics 31(1): 49–57CrossRefMATHGoogle Scholar
  13. Lai TL, Ying ZL (1991) Large sample theory of a modified Buckley–James estimator for regression-analysis with censored-data. Ann Stat 19(3): 1370–1402CrossRefMATHMathSciNetGoogle Scholar
  14. Lai TL, Ying ZL (1992) Linear rank statistics in regression-analysis with censored or truncated data. J Multivar Anal 40(1): 13–45CrossRefMATHMathSciNetGoogle Scholar
  15. Lin DY, Ying ZL (1995) Semiparametric inference for the accelerated life model with time-dependent covariates. J Stat Plan Inference 44(1): 47–63CrossRefMATHMathSciNetGoogle Scholar
  16. Meier P (1975) Estimation of a distribution function form incomplete observations. In: Gani J (eds) Perspectives in probability and statistics. Academic Press, London, pp 67–87Google Scholar
  17. Miller R, Halpern J (1982) Regression with censored-data. Biometrika 69(3): 521–531CrossRefMATHMathSciNetGoogle Scholar
  18. Muller HG, Zhao PL (1995) On a semiparametric variance function model and a test for heteroscedasticity. Ann Stat 23(3): 946–967CrossRefMathSciNetGoogle Scholar
  19. Ritov Y (1990) Estimation in a linear-regression model with censored-data. Ann Stat 18(1): 303–328CrossRefMATHMathSciNetGoogle Scholar
  20. Robins J, Tsiatis AA (1992) Semiparametric estimation of an accelerated failure time model with time-dependent covariates. Biometrika 79(2): 311–319MATHMathSciNetGoogle Scholar
  21. Ruppert D, Wand MP (1994) Multivariate locally weighted least-squares regression. Ann Stat 22(3): 1346–1370CrossRefMATHMathSciNetGoogle Scholar
  22. Yu MG, Nan B (2006) A hybrid Newton-type method for censored survival data using double weights in linear models. Lifetime Data Anal 12(3): 345–364CrossRefMathSciNetGoogle Scholar
  23. Yu L, Yu R, Liu L (2009) Quasi-likelihood for right-censored data in the generalized linear model. Commun Stat Theory Methods 38(13): 2187–2200CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Jiann-Ping Hsu college of Public HealthGeorgia Southern UniversityStatesboroUSA

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