The Journal of Economic Inequality

, Volume 6, Issue 1, pp 73–87 | Cite as

The empirical assessment of multidimensional welfare, inequality and poverty: Sample weighted multivariate generalizations of the Kolmogorov–Smirnov two sample tests for stochastic dominance



Sample weighted multidimensional extensions to existing stochastic dominance, inequality and polarization comparison techniques are introduced and employed to examine whether or not ignoring multidimensional and sample weighting aspects result in misleading inferences. The techniques are employed in the context of a sample of nations, in essence each country in the sample is represented by an agent characterized by the per capita GNP of that country, the GNP growth rate of that country and the average life expectancy in that country. In essence the inequality that is being examined is that between the representative agents in these countries, intra country inequality is not being measured. The results suggest that multidimensional techniques lead to substantially different conclusions from those drawn from the use of unidimensional measures and that sample weighting also has a profound effect on the empirical outcomes.

Key words

welfare inequality poverty 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anderson, G.J.: Non-parametric tests of stochastic dominance in income distributions. Econometrica 64, 1183–1193 (1996)CrossRefGoogle Scholar
  2. 2.
    Anderson, G.J.: A note on the inconsistency of tests employing point-wise comparisons for the equality of two functions. Mimeo, Economics Department University of Toronto (2001a)Google Scholar
  3. 3.
    Anderson, G.J.: The power and size of nonparametric tests for common distributional characteristics. Econom. Rev. 20, 1–30 (2001b)CrossRefGoogle Scholar
  4. 4.
    Anderson, G.J.: Towards an empirical analysis of polarization. J. Econom. 122, 1–26 (2004a)CrossRefGoogle Scholar
  5. 5.
    Anderson, G.J.: Making inferences about the polarization, welfare and poverty of nations: a study of 101 countries 1970–1995. J. Appl. Econ. 19, 537–550 (2004b)CrossRefGoogle Scholar
  6. 6.
    Andrews, D.W.K.: A conditional Kolmogorov test. Econometrica 65, 1097–1128 (1997)CrossRefGoogle Scholar
  7. 7.
    Atkinson, A.B.: The Economics of Inequality, 2nd edn. Clarendon, Oxford (1983)Google Scholar
  8. 8.
    Atkinson, A.B.: On the measurement of inequality. J. Econ. Theory 2, 244–263 (1970)CrossRefGoogle Scholar
  9. 9.
    Atkinson, A.B.: On the measurement of poverty. Econometrica 55, 749–764 (1987)CrossRefGoogle Scholar
  10. 10.
    Atkinson, A.B., Bourguignon, F.: The comparison of multi-dimensioned distributions of economic status. Rev. Econ. Stud. 49, 183–201 (1982)CrossRefGoogle Scholar
  11. 11.
    Barrett, G., Donald, S.: Consistent tests for stochastic dominance. Econometrica 7, 171–103 (2003)Google Scholar
  12. 12.
    Beach, C.M., Davidson, R.: Unrestricted statistical inference with Lorenz Curves and income shares. Rev. Econ. Stud. 50, 723–735 (1983)CrossRefGoogle Scholar
  13. 13.
    Boero, G., Smith, J., Wallis, K.F.: Sensitivity of the chi-squared goodness of fit test to the partitioning of data. Mimeo, Economics Department University of Toronto (2004)Google Scholar
  14. 14.
    Browning, M., Lusardi, A.: Household saving: micro theories and micro facts. J. Econ. Lit. 34, 1797–1855 (1996)Google Scholar
  15. 15.
    Crawford, I.: Nonparametric tests of stochastic dominance in bivariate distributions, with an application to UK data. Institute of Fiscal Studies (1999)Google Scholar
  16. 16.
    Davidson, R., Duclos, J.-Y.: Statistical inference for the measurement of the incidence of taxes and transfers. Econometrica 65, 1453–1466 (1997)CrossRefGoogle Scholar
  17. 17.
    Davidson, R., Duclos, J.-Y.: Statistical inference for stochastic dominance and for the measurement of poverty and inequality. Econometrica 68, 1435–1464 (2000)CrossRefGoogle Scholar
  18. 18.
    Duclos, J.-Y., Sahn, D., Younger, S.: Robust multi-dimensional poverty comparisons. Mimeo, Cornell University (2001)Google Scholar
  19. 19.
    Dvoretzky, A., Kiefer, J., Wolfowitz, J.: Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Ann. Math. Stat. 27, 352–374 (1956)CrossRefGoogle Scholar
  20. 20.
    Esteban, J.-M., Ray, D.: On the measurement of polarization. Econometrica 62, 819–851 (1994)CrossRefGoogle Scholar
  21. 21.
    Formby, J.P., Smith, W.J., Zheng, B., Chow, V.: Inequality orderings, normalized stochastic dominance and statistical inference. J. Bus. Econ. Stat. 18, 479–488 (2000)CrossRefGoogle Scholar
  22. 22.
    Foster, J.E., Shorrocks, A.F.: Poverty orderings. Econometrica 56, 173–177 (1988)CrossRefGoogle Scholar
  23. 23.
    Friedman, M.: A Theory of the Consumption Function. Princeton University Press, Princeton, NJ (1957)Google Scholar
  24. 24.
    Ibbott, P.: A test for stochastic dominance in bivariate distributions with an application to canadian living standards. Mimeo, Department of Economics, Business and Mathematics Kings College University of Western Ontario (2004)Google Scholar
  25. 25.
    Kiefer, J.: On large deviations of the empiric D.F. of vector chance variables and a law of the iterated logarithm. Pac. J. Math. 11, 649–660 (1961)Google Scholar
  26. 26.
    Kiefer, J., Wolfowitz, J.: On the deviations of the empiric distribution function of vector chance variables. Trans. Am. Math. Soc. 87, 173–186 (1958)CrossRefGoogle Scholar
  27. 27.
    Kim, P.J., Jennrich: Tables of the exact sampling distribution of the two sample Kolmogorov-Smirnov criterion D m,n(m ≤ n) 79–170. In: Institute of Mathematical Statistics (ed.) Selected Tables in Mathematical Statistics, vol I. American Mathematical Society, Providence, RI (1973)Google Scholar
  28. 28.
    Koshevoy, G.A., Mosler, K.: Multivariate Gini indices. J. Multivar. Anal. 60, 252–276 (1997)CrossRefGoogle Scholar
  29. 29.
    Linton, O., Maasoumi, E., Whang, Y.-J.: Consistent tests for stochastic dominance: A subsampling approach. Mimeo, LSE (2002)Google Scholar
  30. 30.
    McFadden, D.: Testing for stochastic dominance. In: Fomby, T., Seo, T.K. (eds.) Studies in the Economics of Uncertainty: In Honor of Josef Hadar. Springer, Berlin Heidelberg New York (1989)Google Scholar
  31. 31.
    Maasoumi, E.: The measurement and decomposition of multidimensional inequality. Econometrica 54, 771–779 (1986)CrossRefGoogle Scholar
  32. 32.
    Maasoumi, E.: A compendium of information theory in economics and econometrics. Econom. Rev. 12, 1–49 (1993)Google Scholar
  33. 33.
    Maasoumi, E.: Multidimensioned approaches to welfare analysis. In: Silber, J. (ed.) Chapter 15, Handbook of Income Inequality Measurement. Kluwer, Boston (1999)Google Scholar
  34. 34.
    Modigliani, F., Brumberg, R.: Utility analysis and the consumption function: An interpretation of cross section data. In: Kurihara, K.K. (ed.) Post Keynesian Economics. Rutgers University Press, New Brunswick, NJ (1954)Google Scholar
  35. 35.
    Phelps, E.S.: The golden rule of accumulation: a fable for growthmen. Am. Econ. Rev. 51, 638–643 (1961)Google Scholar
  36. 36.
    Rao, R.C.: Linear Statistical Inference and Its Applications. Wiley, New York (1973)Google Scholar
  37. 37.
    Sen, A.: Inequality Reexamined. Harvard University Press, Cambridge, MA (1995)Google Scholar
  38. 38.
    Tsui, K.-Y.: Multidimensional generalizations of the relative and absolute inequality indices: The Atkinson–Kolm–Sen approach. J. Econ. Theory 67, 251–265 (1995)CrossRefGoogle Scholar
  39. 39.
    Wolak, F.A.: Testing inequality constraints in linear econometric models. J. Econ. 41, 205–235 (1989)Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of TorontoTorontoCanada

Personalised recommendations