New estimating equation approaches with application in lifetime data analysis



Estimating equation approaches have been widely used in statistics inference. Important examples of estimating equations are the likelihood equations. Since its introduction by Sir R. A. Fisher almost a century ago, maximum likelihood estimation (MLE) is still the most popular estimation method used for fitting probability distribution to data, including fitting lifetime distributions with censored data. However, MLE may produce substantial bias and even fail to obtain valid confidence intervals when data size is not large enough or there is censoring data. In this paper, based on nonlinear combinations of order statistics, we propose new estimation equation approaches for a class of probability distributions, which are particularly effective for skewed distributions with small sample sizes and censored data. The proposed approaches may possess a number of attractive properties such as consistency, sufficiency and uniqueness. Asymptotic normality of these new estimators is derived. The construction of new estimation equations and their numerical performance under different censored schemes are detailed via Weibull distribution and generalized exponential distribution.


Estimation equation Nonlinear combination of order statistics Asymptotic normality Weibull distribution Generalized exponential distribution 


  1. Brown, B. M. (1985). Grouping problems in distribution-free regression. Australian Journal of Statistics, 27, 123–134.MathSciNetCrossRefGoogle Scholar
  2. Csörgő, M., Horváth, L. (1990). On the distribution of \(L_p\) norms of weighted quantile processes. Annales de l’I.H.P., section B, 26, 65–85.Google Scholar
  3. Csörgő, M., Révész, P. (1978). Strong approximations of the quantile process. Annals of Statistics, 6, 882–894.Google Scholar
  4. Csörgő, M., Yu, H. (1997). Estimation of total time on test transforms for stationary observations. Stochastic Processes and their Applications, 68, 229–253.Google Scholar
  5. Dodson, B. (1994). Weibull analysis: with software. Milwaukee: ASQ Quality Press.Google Scholar
  6. Giorgi, G. M. (1999). Income inequality measurement: the statistical approach. In J. Silber (Ed.), Handbook of income inequality measurement (pp. 245–267). Boston: Kluwer.CrossRefGoogle Scholar
  7. Gupta, R. D., Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41, 173–188.Google Scholar
  8. Gupta, R. D., Kundu, D. (2001). Generalized exponential distributions: different methods of estimations. Journal of Statistical Computation and Simulation, 69, 315–338.Google Scholar
  9. Gupta, R. D., Kundu, D. (2003). Closeness of gamma and generalized exponential distribution. Communications in Statistics—Theory and Methods, 32, 705–721.Google Scholar
  10. Gupta, R. D., Kundu, D. (2006). On the comparison of Fisher information of the Weibull and GE distributions. Journal of Statistical Planning and Inference, 136, 3130–3144.Google Scholar
  11. Gupta, R. D., Kundu, D. (2007). Generalized exponential distribution: existing results and some recent developments. Journal of Statistical Planning and Inference, 137, 3537–3547.Google Scholar
  12. Hampel, F. R., Rousseeuw, P. J., Ronchetti, E. M., Stahel, W. (1986). Robust statistics: the approach based on influence functions. New York: Wiley Interscience.Google Scholar
  13. Hosking, J. R. M. (1990). L-moments: analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society B, 52, 105–124.MathSciNetMATHGoogle Scholar
  14. Hosking, J. R. M. (1995). The use of L-moments in the analysis of censored data. In N. Balakrishnan (Ed.), Recent advances in life-testing and reliability (pp. 545–564). Boca Raton: CRC Press.Google Scholar
  15. Jones, B. L., Zitikis, R. (2003). Empirical estimation of risk measures and related quantities. North American Actuarial Journal, 7, 44–54.Google Scholar
  16. Kundu, D., Gupta, R. D., Manglick, A. (2005). Discriminating between the log-normal and generalized exponential distribution. Journal of Statistical Planning and Inference, 127, 213–227.Google Scholar
  17. Lawless, J. F. (1975). Construction of tolerance bounds for the extreme value and Weibull distributions. Technometrics, 17, 255–261.MathSciNetMATHCrossRefGoogle Scholar
  18. Lawless, J. F. (2003). Statistical models and methods for lifetime data (2nd ed., pp. 1691–1696). New York: Wiley.MATHGoogle Scholar
  19. Maritz, J. S. (1995). Distribution-free statistical methods. London: Chapman& Hall.Google Scholar
  20. Mitra, S., Kundu, D. (2008). Analysis of the left censored data from the generalized exponential distribution. Journal of Statistical Computation and Simulation, 78, 669–679.Google Scholar
  21. Raqab, M. Z. (2002). Inferences for generalized exponential distribution based on record statistics. Journal of Statistical Planning and Inference, 104, 339–350.MathSciNetMATHCrossRefGoogle Scholar
  22. Wang, B. X., Yu, K., Jones, M. C. (2010). Inference under progressively type II right censored sampling for certain lifetime distributions. Technometrics, 52, 453–460.Google Scholar
  23. Weerahandi, S. (2004). Generalized inference in repeated measures: exact methods in MANOVA and mixed models. New Jersey: Wiley.MATHGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 2012

Authors and Affiliations

  1. 1.Department of Mathematical SciencesBrunel UniversityLondonUK
  2. 2.Business SchoolShihezi UniversityWujiaquChina
  3. 3.Department of StatisticsZhejiang Gongshang UniversityHangzhouChina
  4. 4.Ecole Nationale de la Statistique et de l’Analyse de l’Information (Ensai)(CRES)BRUZ CedexFrance

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