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Lifetime Data Analysis

, Volume 23, Issue 2, pp 275–304 | Cite as

Gini index estimation for lifetime data

Article

Abstract

Lifetime data is often right-censored. Recent literature on the Gini index estimation with censored data focuses on independent censoring. However, the censoring mechanism is likely to be dependent censoring in practice. This paper proposes two estimators of the Gini index under independent censoring and covariate-dependent censoring, respectively. The proposed estimators are consistent and asymptotically normal. We also evaluate the performance of our estimators in finite samples through Monte Carlo simulations. Finally, the proposed methods are applied to real data.

Keywords

Gini index Lifetime data Independent censoring  Covariate-dependent censoring 

Notes

Acknowledgments

The authors are grateful to the Editor, the Associate Editor, two anonymous Referees for their critical and insightful comments, which led to great improvements in the revised manuscript. This work was supported by the National Natural Science Foundation of China (No. 71501159 and 71401112) and the Fundamental Research Funds for the Central Universities (JBK160113).

References

  1. Aalen OO (1980) A model for nonparametric regression analysis of counting processes. In: Mathematical statistics and probability theory. Springer, New YorkGoogle Scholar
  2. Aalen OO (1989) A linear regression model for the analysis of life times. Stat Med 8(8):907–925CrossRefGoogle Scholar
  3. Aalen OO (1993) Further results on the non-parametric linear regression model in survival analysis. Stat Med 12(17):1569–1588CrossRefGoogle Scholar
  4. Andersen P, Borgan O, Gill R, Keiding N (1993) Statistical models based on counting processes. Springer, New YorkCrossRefMATHGoogle Scholar
  5. Beran R (1981) Nonparametric regression with randomly censored survival data. Tech. rep., Technical Report, Univ. California, BerkeleyGoogle Scholar
  6. Berrebi ZM, Silber J (1985) The Gini coefficient and negative income: a comment. Oxford Econ Pap 37(3):525–526CrossRefGoogle Scholar
  7. Bhattacharya D (2007) Inference on inequality from household survey data. J Econo 137(2):674–707MathSciNetCrossRefMATHGoogle Scholar
  8. Bonetti M, Gigliarano C, Muliere P (2009) The Gini concentration test for survival data. Lifetime Data Anal 15(4):493–518MathSciNetCrossRefMATHGoogle Scholar
  9. Ceriani L, Verme P (2012) The origins of the Gini index: extracts from variabilità e mutabilità (1912) by Corrado Gini. J Econ Inequal 10(3):421–443CrossRefGoogle Scholar
  10. Chen CN, Tsaur TW, Rhai TS (1982) The Gini coefficient and negative income. Oxford Econ Pap 34(3):473–478CrossRefGoogle Scholar
  11. Dabrowska DM (1989) Uniform consistency of the kernel conditional Kaplan–Meier estimate. Ann Stat 17(3):1157–1167MathSciNetCrossRefMATHGoogle Scholar
  12. Datta S, Satten GA (2002) Estimation of integrated transition hazards and stage occupation probabilities for non-Markov systems under dependent censoring. Biometrics 58(4):792–802MathSciNetCrossRefMATHGoogle Scholar
  13. David H (1968) Miscellanea: Gini’s mean difference rediscovered. Biometrika 55(3):573–575MATHGoogle Scholar
  14. Davidson R (2009) Reliable inference for the Gini index. J Econom 150(1):30–40MathSciNetCrossRefMATHGoogle Scholar
  15. Fleming T, Harrington D (1991) Counting processes and survival analysis. Wiley, New YorkMATHGoogle Scholar
  16. Gastwirth JL (1972) The estimation of the Lorenz curve and Gini index. Rev Econ Stat 54(3):306–316MathSciNetCrossRefGoogle Scholar
  17. Gigliarano C, Muliere P (2013) Estimating the Lorenz curve and Gini index with right censored data: a polya tree approach. Metron 71(2):105–122MathSciNetCrossRefMATHGoogle Scholar
  18. Gill RD (1980) Censoring and stochastic integrals. Stat Neerl 34(2):124–124CrossRefMATHGoogle Scholar
  19. Gini C (1912) Variabilità e mutabilità. Reprinted in Memorie di metodologica statistica (Ed Pizetti E, Salvemini, T) Rome: Libreria Eredi Virgilio Veschi 1Google Scholar
  20. Hanada K (1983) A formula of Gini’s concentration ratio and its application to life tables. J Jpn Stat Soc 13(2):95–98MATHGoogle Scholar
  21. Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 19(3):293–325MathSciNetCrossRefMATHGoogle Scholar
  22. Kendall M, Stuart A (1977) The advanced theory of statistics, vol 1., Distribution theoryMacmillan, New YorkMATHGoogle Scholar
  23. Lambert PJ, Aronson JR (1993) Inequality decomposition analysis and the Gini coefficient revisited. Econ J 103(420):1221–1227CrossRefGoogle Scholar
  24. Langel M, Tillé Y (2013) Variance estimation of the Gini index: revisiting a result several times published. J Roy Stat Soc A Sta 176(2):521–540MathSciNetCrossRefGoogle Scholar
  25. Leconte E, Poiraud-Casanova S, Thomas-Agnan C (2002) Smooth conditional distribution function and quantiles under random censorship. Lifetime Data Anal 8(3):229–246MathSciNetCrossRefMATHGoogle Scholar
  26. Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 9(70):209–219Google Scholar
  27. Lubrano M (2012) The econometrics of inequality and poverty. Lecture 4: Lorenz curves, the Gini Coefficient and parametric distributionsGoogle Scholar
  28. Martinussen T, Scheike TH (2006) Dynamic regression models for survival data. Springer, New YorkMATHGoogle Scholar
  29. McCall BP (1996) Unemployment insurance rules, joblessness, and part-time work. Econometrica 64(3):647–682CrossRefMATHGoogle Scholar
  30. Michetti B, Dall’Aglio G (1957) La differenza semplice media. Statistica 7(2):159–255MathSciNetGoogle Scholar
  31. Ogwang T (2000) A convenient method of computing the Gini index and its standard error. Oxford B Econ Stat 62(1):123–129CrossRefGoogle Scholar
  32. Peng L (2011) Empirical likelihood methods for the Gini index. Aust Nz J Stat 53(2):131–139MathSciNetCrossRefMATHGoogle Scholar
  33. Qin Y, Rao J, Wu C (2010) Empirical likelihood confidence intervals for the Gini measure of income inequality. Econ Model 27(6):1429–1435CrossRefGoogle Scholar
  34. Raffinetti E, Siletti E, Vernizzi A (2015) On the Gini coefficient normalization when attributes with negative values are considered. Stat Method Appl 24(3):507–521MathSciNetCrossRefMATHGoogle Scholar
  35. Robins JM, Rotnitzky A (1992) Recovery of information and adjustment for dependent censoring using surrogate markers. In: AIDS Epidemiology (pp. 297–331). Birkhäuser, BostonGoogle Scholar
  36. Robins JM, Rotnitzky A (2005) Inverse probability weighted estimation in survival analysis. In: Encyclopedia of Biostatistics (pp. 2619–2625). Wiley, New YorkGoogle Scholar
  37. Satten GA, Datta S, Robins JM (2001) An estimator for the survival function when data are subject to dependent censoring. Stat Probab Lett 54:397–403CrossRefMATHGoogle Scholar
  38. Scharfstein DO, Rotnitzky A, Robins JM (1999) Adjusting for nonignorable drop-out using semiparametric nonresponse models. J Am Stat Assoc 94(448):1096–1120MathSciNetCrossRefMATHGoogle Scholar
  39. Sen A (1973) On economic inequality. Oxford University Press, OxfordCrossRefGoogle Scholar
  40. Sendler W (1979) On statistical inference in concentration measurement. Metrika 26(1):109–122MathSciNetCrossRefMATHGoogle Scholar
  41. Sengupta M (2009) Unemployment duration and the measurement of unemployment. J Econ Inequal 7(3):273–294CrossRefGoogle Scholar
  42. Shorrocks AF (1980) The class of additively decomposable inequality measures. Econometrica 48(3):613–625MathSciNetCrossRefMATHGoogle Scholar
  43. Sun L, Song X, Zhang Z (2012) Mean residual life models with time-dependent coefficients under right censoring. Biometrika 99(1):185–197MathSciNetCrossRefMATHGoogle Scholar
  44. Sun Y, Lee J (2011) Testing independent censoring for longitudinal data. Stat Sinica 21(3):1315MathSciNetCrossRefMATHGoogle Scholar
  45. Tse SM (2006) Lorenz curve for truncated and censored data. Ann Inst Stat Math 58(4):675–686MathSciNetCrossRefMATHGoogle Scholar
  46. Xu K (2003) How has the literature on Gini’s index evolved in the past 80 years? Dalhousie University, Economics Working PaperGoogle Scholar
  47. Yitzhaki S (1991) Calculating jackknife variance estimators for parameters of the Gini method. J Bus Econ Stat 9(2):235–239Google Scholar
  48. Yitzhaki S, Schechtman E (2013) The Gini methodology: a primer on a statistical methodology. Springer, New YorkCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of International BusinessSouthwestern University of Finance and EconomicsChengduChina
  2. 2.School of Public Policy and ManagementUniversity of Chinese Academy of ScienceBeijingChina
  3. 3.Institute of Policy and ManagementChinese Academy of ScienceBeijingChina
  4. 4.School of EconomicsCapital University of Economics and BusinessBeijingChina

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