We intend to point out the differences which exist
between the classical Gini concentration coefficient and a proposed
bipolarization index defined for an arbitrary random variable which
have a finite support.
In fact Gini's index measures only the "poverty degree" for the
individuals from a given population taking into consideration their
wages. The Gini coefficient is not so sensitive to the significant
income variations in the "rich people class" .
In practice there are multiple interdependent relations between the
pauperization and the socio-economical polarization phenomena. The
presence of a strong pauperization aspect inside the population
induces often a polarization effect in this society. But the
pauperization and the polarization phenomena are not identical. For
this reason it isn't always adequate to use a Gini type coefficient,
based on the Lorenz order, to estimate the bipolarization level of the
individuals from the studied population.
The present paper emphasizes these ideas by considering two
families of random variables which have a linear or a triangular type
distributions. In addition, the continuous variation, depending on the
parameter "time" of the chosen distributions, could simulate a real
dynamical evolution of the population.
[1] A. Agresti, An introduction to categorical data analysis. New York:
Wiley Series in Probability and Statistics, 1996.
[2] C. D'Ambrosio, "Household characteristics and the distribution of
income in Italy - An application of social distance measures," The
Review of Income and Wealth, vol. 47, no. 1, pp. 43-64, 2001.
[3] A. B. Atkinson, "On the measurement of inequality." Journal of
Economic Theory, no. 2, pp. 244-263, 1970.
[4] D. J. Bartolomew, The statistical approach to social measurement.
London: Academic Press, 1996.
[5] N. Bhattacharya, and B. Mahalanobis, "Regional disparities in
household consumption in India," Journal of the American Statistical
Association, vol. 62, pp. 143-161, 1967.
[6] F. Bourguignon, "Decomposable income inequality measures,"
Econometrica, vol. 47, pp. 901-920, 1979.
[7] A. Chateauneuf, T. Gajdos, and P. H. Wilthien, "The principle of strong
diminishing transfer," Journal of Economic Theory, vol. 103, pp. 311-
333, 2002.
[8] F. A. Cowell, "On the structure of additive inequality measures," Review
of Economic Studies, vol. 47, pp. 521-531, 1980.
[9] F. A. Cowell, and S. P. Jenkins. "How much inequality can we explain ?
A methodology and an application to the United States," The Economic
Journal, vol. 105, pp. 421-430, 1995.
[10] J. B. Davies, and A. Shorrocks, "Optimal grouping of income and wealth
data," Journal of Econometrics, vol. 42, pp. 97-108, 1989.
[11] J. Esteban, and D. Ray, "On the measurement of polarization,"
Econometrica, vol. 62 , no. 4, pp. 819-852, 1994.
[12] J. Esteban, and D. Ray, "Conflict and distribution," Journal of Economic
Theory, vol. 87, pp. 379-415, 1999.
[13] J. Esteban, C. Gradin, and D. Ray, "Extension of a measure of
polarization with application to the income distribution of five OECD
countries," Maxwell School of Citizenship and Public Affairs, Syracuse
University, New York, Working paper no. 218, November 1999.
[14] B. S. Everitt, S. Landau, and M. Leese, Cluster analysis. London:
Arnold, 2001.
[15] J. Foster, and A. K. Sen, On economic inequality. Oxford: Clarendon
Press, 1997.
[16] J. E. Foster, and A. A. Shneyerov, "Path independent inequality
measures," Journal of Economic Theory, Oxford, vol. 91, no. 2, pp. 199-
222, 2000.
[17] M. Fournier, "Inequality decomposition by factor component - A new
approach illustrated on the Taiwanese case," CERDI, Université
d-Auvergne, November 1999.
[18] J. Gastwirth, "The estimation of a family of measures of economic
inequality," Journal of Econometrics, no. 3, pp. 61-70, 1975.
[19] C. Gradin, "Polarization by sub-populations in Spain, 1973-1991,"
Review of Income and Wealth, vol. 46, no. 4, pp. 457-474, December
2000.
[20] P. E. Hart, "The comparative statistics and dynamics of income
distributions," Journal of the Royal Statistical Society, vol. 139, pp. 108-
125, 1976.
[21] N. C. Kakwani, "Statistical inference in the measurement of poverty,"
Review of Economics and Statistics, vol. 75, no. 3, pp. 632-639, 1993.
[22] C. Kleiber, and S. Kotz, Statistical size distributions in economics and
actuarial sciences. New Jersey: John Wiley & Sons - Interscience, 2003.
[23] J. O. Lanjouw, and P. Lanjouw, "How to compare apples and oranges -
Poverty measurement based on different definitions of consumption,"
Review of Income and Wealth, vol. 47, no. 1, pp. 25-42, 2001.
[24] A. W. Pedersen, "Inequality as relative deprivation - Theoretical issues
and implications for empirical research," preprint ISA RC19
Conference 2001, "Old a new social inequalities", University of Oviedo,
September 2001.
[25] G. Pyatt, "On the interpretation and disaggregation of Gini coefficient,"
The Economic Journal, vol. 86, pp. 243-255, 1976.
[26] F. Schmid, "A general class of poverty measures," Statistical Papers,
vol. 34, pp. 189-211, 1993.
[27] A. K. Sen, On economic inequality. Oxford: Oxford University Press,
1973.
[28] A. F. Shorrocks, "The class additively decomposable inequality
measures," Econometrica, vol. 48, pp. 613-625, 1980.
[29] A. F. Shorrocks, "Inequality decomposition by population subgroups,"
Econometrica, vol. 52, pp. 1369-1385, 1984.
[30] P. Stefanescu, and S. Stefanescu, "Extending the Gini index to
measure inequality and poverty," Economic Computation and
Economic Cybernetics Studies and Research, vol. 35, no. 1-4, pp. 145-
154, 2001.
[31] P. Stefanescu, and S. Stefanescu, "Comparing Gini index with three
other concentration measures," Economic Computation and Economic
Cybernetics Studies and Research, vol. 36, no. 1-4, pp. 173-184, 2002.
[32] P. Stefanescu, and S. Stefanescu, "The polarization index for bounded
exponential distributions," Economic Computation and Economic
Cybernetics Studies and Research, vol. 40, no. 3-4, pp. 211-218, 2006.
[33] P. Stefanescu, and S. Stefanescu, "The properties of a polarization index
for bounded exponential distributions," U. P. B. - Scientific Bulletin,
Series A : Applied Mathematics and Physics, vol. 68, no. 4, pp. 9-20,
2006.
[34] P. Stefanescu, and S. Stefanescu, "The Monte Carlo estimation of a
polarization index for an arbitrary distribution," Economic Computation
and Economic Cybernetics Studies and Research, vol. 41, no. 3-4, pp.
181-192, 2007.
[35] S. Stefanescu, "Measuring the socio-economic bipolarization
phenomenon," Romanian Journal of Economic Forecasting, vol. 9, no.
1, pp. 149-161, 2008.
[36] K. Y. Tsui, "Multidimensional inequality and multidimensional entropy
measures - An axiomatic derivation," Social Choice and Welfare, vol.
16, no 1, pp. 145-157, 1999.
[37] B. Wilfling, "Lorenz ordering of power function order statistics,"
Statistics & Probability Letters, vol. 30, pp. 313-319, 1996.
[38] M. Wolfson, "When inequalities diverge," American Economic Review,
vol. 84, no. 2, pp. 353-358, 1994.
[39] M. Wolfon, "Divergent inequalities - Theory and empirical results," The
Review of Income and Wealth, vol. 43, no. 4, pp. 401-422, 1997.
[40] S. Yitzhaki, "Economic distance and overlapping of distributions,"
Journal of Econometrics, vol. 61, pp. 147-159, 1994.
[41] X. Zhang, and R. Kanbur, "What difference do polarization measures
make ? - An application to China," Journal of Development Studies, vol.
37, pp. 85-98, 2001.
[42] C. Zoli, "Intersecting generalized Lorenz curves and the Gini index,"
Social Choice and Welfare, vol. 16, pp. 183-196, 1999.
[1] A. Agresti, An introduction to categorical data analysis. New York:
Wiley Series in Probability and Statistics, 1996.
[2] C. D'Ambrosio, "Household characteristics and the distribution of
income in Italy - An application of social distance measures," The
Review of Income and Wealth, vol. 47, no. 1, pp. 43-64, 2001.
[3] A. B. Atkinson, "On the measurement of inequality." Journal of
Economic Theory, no. 2, pp. 244-263, 1970.
[4] D. J. Bartolomew, The statistical approach to social measurement.
London: Academic Press, 1996.
[5] N. Bhattacharya, and B. Mahalanobis, "Regional disparities in
household consumption in India," Journal of the American Statistical
Association, vol. 62, pp. 143-161, 1967.
[6] F. Bourguignon, "Decomposable income inequality measures,"
Econometrica, vol. 47, pp. 901-920, 1979.
[7] A. Chateauneuf, T. Gajdos, and P. H. Wilthien, "The principle of strong
diminishing transfer," Journal of Economic Theory, vol. 103, pp. 311-
333, 2002.
[8] F. A. Cowell, "On the structure of additive inequality measures," Review
of Economic Studies, vol. 47, pp. 521-531, 1980.
[9] F. A. Cowell, and S. P. Jenkins. "How much inequality can we explain ?
A methodology and an application to the United States," The Economic
Journal, vol. 105, pp. 421-430, 1995.
[10] J. B. Davies, and A. Shorrocks, "Optimal grouping of income and wealth
data," Journal of Econometrics, vol. 42, pp. 97-108, 1989.
[11] J. Esteban, and D. Ray, "On the measurement of polarization,"
Econometrica, vol. 62 , no. 4, pp. 819-852, 1994.
[12] J. Esteban, and D. Ray, "Conflict and distribution," Journal of Economic
Theory, vol. 87, pp. 379-415, 1999.
[13] J. Esteban, C. Gradin, and D. Ray, "Extension of a measure of
polarization with application to the income distribution of five OECD
countries," Maxwell School of Citizenship and Public Affairs, Syracuse
University, New York, Working paper no. 218, November 1999.
[14] B. S. Everitt, S. Landau, and M. Leese, Cluster analysis. London:
Arnold, 2001.
[15] J. Foster, and A. K. Sen, On economic inequality. Oxford: Clarendon
Press, 1997.
[16] J. E. Foster, and A. A. Shneyerov, "Path independent inequality
measures," Journal of Economic Theory, Oxford, vol. 91, no. 2, pp. 199-
222, 2000.
[17] M. Fournier, "Inequality decomposition by factor component - A new
approach illustrated on the Taiwanese case," CERDI, Université
d-Auvergne, November 1999.
[18] J. Gastwirth, "The estimation of a family of measures of economic
inequality," Journal of Econometrics, no. 3, pp. 61-70, 1975.
[19] C. Gradin, "Polarization by sub-populations in Spain, 1973-1991,"
Review of Income and Wealth, vol. 46, no. 4, pp. 457-474, December
2000.
[20] P. E. Hart, "The comparative statistics and dynamics of income
distributions," Journal of the Royal Statistical Society, vol. 139, pp. 108-
125, 1976.
[21] N. C. Kakwani, "Statistical inference in the measurement of poverty,"
Review of Economics and Statistics, vol. 75, no. 3, pp. 632-639, 1993.
[22] C. Kleiber, and S. Kotz, Statistical size distributions in economics and
actuarial sciences. New Jersey: John Wiley & Sons - Interscience, 2003.
[23] J. O. Lanjouw, and P. Lanjouw, "How to compare apples and oranges -
Poverty measurement based on different definitions of consumption,"
Review of Income and Wealth, vol. 47, no. 1, pp. 25-42, 2001.
[24] A. W. Pedersen, "Inequality as relative deprivation - Theoretical issues
and implications for empirical research," preprint ISA RC19
Conference 2001, "Old a new social inequalities", University of Oviedo,
September 2001.
[25] G. Pyatt, "On the interpretation and disaggregation of Gini coefficient,"
The Economic Journal, vol. 86, pp. 243-255, 1976.
[26] F. Schmid, "A general class of poverty measures," Statistical Papers,
vol. 34, pp. 189-211, 1993.
[27] A. K. Sen, On economic inequality. Oxford: Oxford University Press,
1973.
[28] A. F. Shorrocks, "The class additively decomposable inequality
measures," Econometrica, vol. 48, pp. 613-625, 1980.
[29] A. F. Shorrocks, "Inequality decomposition by population subgroups,"
Econometrica, vol. 52, pp. 1369-1385, 1984.
[30] P. Stefanescu, and S. Stefanescu, "Extending the Gini index to
measure inequality and poverty," Economic Computation and
Economic Cybernetics Studies and Research, vol. 35, no. 1-4, pp. 145-
154, 2001.
[31] P. Stefanescu, and S. Stefanescu, "Comparing Gini index with three
other concentration measures," Economic Computation and Economic
Cybernetics Studies and Research, vol. 36, no. 1-4, pp. 173-184, 2002.
[32] P. Stefanescu, and S. Stefanescu, "The polarization index for bounded
exponential distributions," Economic Computation and Economic
Cybernetics Studies and Research, vol. 40, no. 3-4, pp. 211-218, 2006.
[33] P. Stefanescu, and S. Stefanescu, "The properties of a polarization index
for bounded exponential distributions," U. P. B. - Scientific Bulletin,
Series A : Applied Mathematics and Physics, vol. 68, no. 4, pp. 9-20,
2006.
[34] P. Stefanescu, and S. Stefanescu, "The Monte Carlo estimation of a
polarization index for an arbitrary distribution," Economic Computation
and Economic Cybernetics Studies and Research, vol. 41, no. 3-4, pp.
181-192, 2007.
[35] S. Stefanescu, "Measuring the socio-economic bipolarization
phenomenon," Romanian Journal of Economic Forecasting, vol. 9, no.
1, pp. 149-161, 2008.
[36] K. Y. Tsui, "Multidimensional inequality and multidimensional entropy
measures - An axiomatic derivation," Social Choice and Welfare, vol.
16, no 1, pp. 145-157, 1999.
[37] B. Wilfling, "Lorenz ordering of power function order statistics,"
Statistics & Probability Letters, vol. 30, pp. 313-319, 1996.
[38] M. Wolfson, "When inequalities diverge," American Economic Review,
vol. 84, no. 2, pp. 353-358, 1994.
[39] M. Wolfon, "Divergent inequalities - Theory and empirical results," The
Review of Income and Wealth, vol. 43, no. 4, pp. 401-422, 1997.
[40] S. Yitzhaki, "Economic distance and overlapping of distributions,"
Journal of Econometrics, vol. 61, pp. 147-159, 1994.
[41] X. Zhang, and R. Kanbur, "What difference do polarization measures
make ? - An application to China," Journal of Development Studies, vol.
37, pp. 85-98, 2001.
[42] C. Zoli, "Intersecting generalized Lorenz curves and the Gini index,"
Social Choice and Welfare, vol. 16, pp. 183-196, 1999.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59921", author = "Stefan V. Stefanescu", title = "Measurement of the Bipolarization Events", abstract = "We intend to point out the differences which exist
between the classical Gini concentration coefficient and a proposed
bipolarization index defined for an arbitrary random variable which
have a finite support.
In fact Gini's index measures only the "poverty degree" for the
individuals from a given population taking into consideration their
wages. The Gini coefficient is not so sensitive to the significant
income variations in the "rich people class" .
In practice there are multiple interdependent relations between the
pauperization and the socio-economical polarization phenomena. The
presence of a strong pauperization aspect inside the population
induces often a polarization effect in this society. But the
pauperization and the polarization phenomena are not identical. For
this reason it isn't always adequate to use a Gini type coefficient,
based on the Lorenz order, to estimate the bipolarization level of the
individuals from the studied population.
The present paper emphasizes these ideas by considering two
families of random variables which have a linear or a triangular type
distributions. In addition, the continuous variation, depending on the
parameter "time" of the chosen distributions, could simulate a real
dynamical evolution of the population.", keywords = "Bipolarization phenomenon, Gini coefficient, income distribution, poverty measure.", volume = "3", number = "9", pages = "690-8", }