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Minimum Risk Point Estimation of Gini Index

Abstract

This paper develops a theory and methodology for estimation of Gini index such that both cost of sampling and estimation error are minimum. Methods in which sample size is fixed in advance, cannot minimize estimation error and sampling cost at the same time. In this article, a purely sequential procedure is proposed which provides an estimate of the sample size required to achieve a sufficiently smaller estimation error and lower sampling cost. Characteristics of the purely sequential procedure are examined and asymptotic optimality properties are proved without assuming any specific distribution of the data. Performance of our method is examined through extensive simulation study.

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References

  • Aguirregabiria, V. and Mira, P. (2007). Sequential estimation of dynamic discrete games. Econometrica 75, 1–53.

    MathSciNet  Article  MATH  Google Scholar 

  • Allison, P.D. (1978). Measures of inequality. Am. Sociol. Rev., 865–880.

  • Arcidiacono, P. and Jones, J.B. (2003). Finite mixture distributions, sequential likelihood and the em algorithm. Econometrica 71, 933–946.

    MathSciNet  Article  MATH  Google Scholar 

  • Asada, Y. (2005). Assessment of the health of americans: the average health-related quality of life and its inequality across individuals and groups. Popul. Health Metrics 3, 7.

    Article  Google Scholar 

  • Beach, C.M. and Davidson, R. (1983). Distribution-free statistical inference with lorenz curves and income shares. Rev. Econ. Stud. 50, 723–735.

    Article  MATH  Google Scholar 

  • Chattopadhyay, B. and De, S.K. (2014). Estimation accuracy of an inequality index. Recent advances in applied mathematics, modelling and simulation. In: Mastorakis, N. E., Demiralp, M., Mukhopadhyay, N. and Mainard, F. (eds.) Recent Advances in Applied Mathematics, Modelling and Simulation, WSEAS, p. 318–321.

    Google Scholar 

  • Chattopadhyay, B. and De, S.K. (2016). Estimation of Gini index within pre-specified error bound. Econometrics 4, 30. doi:10.3390/econometrics4030030.

    Article  Google Scholar 

  • Chattopadhyay, B. and Mukhopadhyay, N. (2013). Two-stage fixed-width confidence intervals for a normal mean in the presence of suspect outliers. Seq. Anal. 32, 134–157.

    MathSciNet  Article  MATH  Google Scholar 

  • Cochran, W.G. (1977). Sampling techniques, 98. Wiley, New York.

    MATH  Google Scholar 

  • Dantzig, G.B. (1940). On the non-existence of tests of “student’s” hypothesis having power functions independent of σ. Ann. Math. Stat. 11, 186–192.

    MathSciNet  Article  MATH  Google Scholar 

  • Das, A. and Rout, H.S. (2015). The social sector in India: issues and challenges. Cambridge Scholars Publishing, UK. chap 10.

    Google Scholar 

  • Davidson, R. (2009). Reliable inference for the Gini index. J. Econ. 150, 30–40.

    MathSciNet  Article  MATH  Google Scholar 

  • Davidson, R. and Duclos, J.Y. (2000). Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica 68, 1435–1464.

    MathSciNet  Article  MATH  Google Scholar 

  • Doob, J.L. (1953). Stochastic processes. Wiley, New York.

    MATH  Google Scholar 

  • Gastwirth, J.L. (1972). The estimation of the Lorenz curve and Gini index. Rev. Econ. Stat., 306–316.

  • Ghosh, B.K. and Sen, P.K. (1991). Handbook of sequential analysis, vol 118. CRC Press.

  • Ghosh, M. and Mukhopadhyay, N. (1979). Sequential point estimation of the mean when the distribution is unspecified. Commun. Stat.-Theory Methods 8, 637–652.

    MathSciNet  Article  MATH  Google Scholar 

  • Ghosh, M., Mukhopadhyay, N. and Sen, P.K. (1997). Sequential estimation. Wiley, New York.

    Book  MATH  Google Scholar 

  • Greene, W.H. (1998). Gender economics courses in liberal arts colleges: further results. J. Econ. Educ. 29, 291–300.

    Article  Google Scholar 

  • Gut, A. (2009). Stopped random walks: Limit theorems and applications. Springer.

  • Hoeffding, W. (1948). A class of statistics with asymptotically normal distribution. Ann. Math. Stat. 19, 293–325.

    MathSciNet  Article  MATH  Google Scholar 

  • Hoeffding, W. (1961). The strong law of large numbers for u-statistics. Institute of Statistics mimeo series 302.

  • Hollander, M. and Wolfe, D.A. (1999). Nonparametric statistical methods. Wiley, New York.

    MATH  Google Scholar 

  • Kanninen, B.J. (1993). Design of sequential experiments for contingent valuation studies. J. Environ. Econ. Manag. 25, S1–S11.

    Article  Google Scholar 

  • Lee, A.J. (1990). U-statistics: theory and practice. CRC Press.

  • Loève, M. (1963). Probability theory. Van Nostrand, Princeton.

    MATH  Google Scholar 

  • Loomes, G. and Sugden, R. (1982). Regret theory: an alternative theory of rational choice under uncertainty. Econ. J. 92, 805–824.

    Article  Google Scholar 

  • Mukhopadhyay, N. and Chattopadhyay, B. (2012). A tribute to Frank Anscombe and random central limit theorem from 1952. Seq. Anal. 31, 265–277.

    MathSciNet  MATH  Google Scholar 

  • Mukhopadhyay, N. and De Silva, B.M. (2009). Sequential methods and their applications. CRC Press.

  • Robbins, H. (1959). Sequential estimation of the mean of a normal population. Almquist and Wiksell, Uppsala, p. 235–245.

  • Sen, P.K. (1981). Sequential nonparametrics: invariance principles and statistical inference. Wiley, New York.

    MATH  Google Scholar 

  • Sen, P.K. (1988). Functional jackknifing: rationality and general asymptotics. Ann. Stat., 450–469.

  • Sen, P.K. and Ghosh, M. (1981). Sequential point estimation of estimable parameters based on u-statistics. Sankhyā: The Indian Journal of Statistics, Series A pp. 331–344.

  • Shi, H. and Sethu, H. (2003). Greedy fair queueing: a goal-oriented strategy for fair real-time packet scheduling. IEEE, p. 345–356.

  • Sproule, R. (1969). A sequential fixed-width confidence interval for the mean of a u-statistic. PhD thesis Ph, D. dissertation, Univ. of North Carolina.

  • Thomas, V., Wang, Y. and Fan, X. (2001). Measuring education inequality: Gini coefficients of education, vol. 2525. World Bank Publications.

  • Wittebolle, L., Marzorati, M., Clement, L., Balloi, A., Daffonchio, D., Heylen, K., De Vos, P., Verstraete, W. and Boon, N. (2009). Initial community evenness favours functionality under selective stress. Nature 458, 623–626.

    Article  Google Scholar 

  • Xu, K. (2007). U-statistics and their asymptotic results for some inequality and poverty measures. Econ. Rev. 26, 567–577.

    MathSciNet  Article  MATH  Google Scholar 

  • Yitzhaki, S. and Schechtman, E. (2013). The Gini methodology: a primer on a statistical methodology. Springer.

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Acknowledgements

We are grateful to the associate editor and the two anonymous referees for their valuable feedback that helped us improve this manuscript. We also thank Prof. Amarendra Das for providing the income data, which helped us show an application of the proposed method of this article.

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Correspondence to Shyamal K. De.

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De, S.K., Chattopadhyay, B. Minimum Risk Point Estimation of Gini Index. Sankhya B 79, 247–277 (2017). https://doi.org/10.1007/s13571-017-0140-3

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Keywords and phrases.

  • Asymptotic efficiency
  • Ratio regret
  • Reverse submartingale
  • Sequential point estimation
  • Simple random sampling
  • U-statistics

AMS (2000) subject classification.

  • Primary: 62L12
  • 62G05
  • Secondary: 60G46
  • 60G40
  • 91B82