1. bookVolume 37 (2021): Issue 4 (December 2021)
Journal Details
License
Format
Journal
eISSN
2001-7367
First Published
01 Oct 2013
Publication timeframe
4 times per year
Languages
English
access type Open Access

Robust Estimation of the Theil Index and the Gini Coeffient for Small Areas

Published Online: 26 Dec 2021
Page range: 955 - 979
Received: 01 Oct 2019
Accepted: 01 Jan 2021
Journal Details
License
Format
Journal
eISSN
2001-7367
First Published
01 Oct 2013
Publication timeframe
4 times per year
Languages
English
Abstract

Small area estimation is receiving considerable attention due to the high demand for small area statistics. Small area estimators of means and totals have been widely studied in the literature. Moreover, in the last years also small area estimators of quantiles and poverty indicators have been studied. In contrast, small area estimators of inequality indicators, which are often used in socio-economic studies, have received less attention. In this article, we propose a robust method based on the M-quantile regression model for small area estimation of the Theil index and the Gini coefficient, two popular inequality measures. To estimate the mean squared error a non-parametric bootstrap is adopted. A robust approach is used because often inequality is measured using income or consumption data, which are often non-normal and affected by outliers. The proposed methodology is applied to income data to estimate the Theil index and the Gini coefficient for small domains in Tuscany (provinces by age groups), using survey and Census micro-data as auxiliary variables. In addition, a design-based simulation is carried out to study the behaviour of the proposed robust estimators. The performance of the bootstrap mean squared error estimator is also investigated in the simulation study.

Keywords

Alfons, A. and M. Templ. 2013. “Estimation of social exclusion indicators from complex surveys: The r package laeken.” Journalof Statistical Software 54 (15): 1–25. DOI: https://doi.org/10.18637/jss.v054.i15.10.18637/jss.v054.i15 Search in Google Scholar

Battacharya, D. 2007. “Inference on inequality from household survey data.” Journal of Econometrics 137: 674–707. DOI: https://doi.org/10.1016/j.jeconom.2005.09.003.10.1016/j.jeconom.2005.09.003 Search in Google Scholar

Bianchi, A., E. Fabrizi, N. Salvati, and N. Tzavidis. 2018. “Estimation and testing in m-quantile regression with applications to small area estimation.” International Statistical Review 86 (3): 541–570. DOI: https://doi.org/10.1111/insr.12267.10.1111/insr.12267 Search in Google Scholar

Bourguignon, F. 1979. “Decomposable income inequality measures.” Econometrica 42: 27–41. DOI: https://doi.org/10.2307/1914138.10.2307/1914138 Search in Google Scholar

Box, G., and D. Cox. 1964. “An analysis of transformations.” Journal of the Royal Statistical Society Series B 27 (2): 211–252. DOI: https://doi.org/10.1111/j.2517-6161.1964.tb00553.x.10.1111/j.2517-6161.1964.tb00553.x Search in Google Scholar

Breckling, J., and R. Chambers. 1988. “M-quantiles.” Biometrika 75 (4): 761–771. DOI: https://doi.org/10.1093/biomet/75.4.761.10.1093/biomet/75.4.761 Search in Google Scholar

Chambers, R.L. 1986. “Outlier robust finite population estimation.” Journal of the American Statistical Associationtion 81 (396): 1063–1069. DOI: https://doi.org/10.1111/rssb.12019.10.1111/rssb.12019 Search in Google Scholar

Chambers, R., H. Chandra, N. Salvati, and N. Tzavidis. 2014. “Outlier robust small area estimation.” Journal of the Royal Statistical Society Series B 76 (1): 47–69. DOI: https://doi.org/10.1111/rssb.12019.10.1111/rssb.12019 Search in Google Scholar

Chambers, R., and Dunstan. 1986. “Estimating distribution function from survey data.” Biometrika 73: 597–604. DOI: https://doi.org/10.1093/biomet/73.3.597.10.1093/biomet/73.3.597 Search in Google Scholar

Chambers, R., and N. Tzavidis. 2006. “M-quantile models for small area estimation.” Biometrika 93 (2): 255–268. DOI: https://doi.org/10.1093/biomet/93.2.255.10.1093/biomet/93.2.255 Search in Google Scholar

Cowell, F., and K. Kuga. 1981. “Inequality measurement: An axiomatic approach.” Journal of Economic Theory 15: 287–305. DOI: https://doi.org/10.1016/S0014-2921(81)80003-7.10.1016/S0014-2921(81)80003-7 Search in Google Scholar

Davidson, R. 2009. “Reliable inference for the gini index.” Journal of Econometrics 150: 30–40. DOI: https://doi.org/10.1016/j.jeconom.2008.11.004.10.1016/j.jeconom.2008.11.004 Search in Google Scholar

Davidson, R., and E. Flachaire. 2007. “Asymptotic and bootstrap inference for inequality and poverty measures.” Journal of Econometrics 141 (1): 141–66. DOI: https://doi.org/10.1016/j.jeconom.2007.01.009.10.1016/j.jeconom.2007.01.009 Search in Google Scholar

Deltas, G. 2003. “The small-samples bias of the gini coefficient: results and implications for empirical research.” The Review of Economics and Statistics 85: 226–34. DOI: https://doi.org/10.1162/rest.2003.85.1.226.10.1162/rest.2003.85.1.226 Search in Google Scholar

Diallo, M.S., and J.N.K. Rao. 2018. “Small area estimation of complex parameters under unit-level models with skew-normal errors.” Scandinavian Journal of Statistics 45 (4): 1092–1116. DOI: https://doi.org/10.1111/sjos.12336.10.1111/sjos.12336 Search in Google Scholar

Dongomo-Jiongo, V., and P. Nguimkeu. 2018. Bootstrapping mean squared errors of robust small-area estimators: Application to the method-of-payments data. Technical report, Staff Working Paper: 18–28, Bank of Canada. Available at: https://www.bankofcanada.ca/wp-content/uploads/2018/06/swp2018-28.pdf. (accessed November 2021). Search in Google Scholar

Elbers, C., J.O. Lanjouw, and P. Lanjouw. 2003. “Micro-level estimation of poverty and inequality.” Econometrica 71 (1): 355–364. DOI: https://www.jstor.org/stable/3082050.10.1111/1468-0262.00399 Search in Google Scholar

Elbers, C., and R. van der Weide. 2014. Estimation of Normal Mixtures in a Nested Error Model with an Application to Small Area Estimation of Poverty and Inequality. The World Bank. Available at: https://openknowledge.worldbank.org/handle/10986/19362. (accessed November 2021).10.1596/1813-9450-6962 Search in Google Scholar

Foster, J. 1983. “An axiomatic characteriazation of the Theil measure of income inequality.” Journal of Economic Theory 31: 105–121. DOI: https://doi.org/10.1016/0022-0531(83)90023-6.10.1016/0022-0531(83)90023-6 Search in Google Scholar

Foster, J., J. Greer, and E. Thorbecke. 1984. “A class of decomposable poverty measures.” Econometrica 52: 761–766. DOI: https://doi.org/10.2307/1913475.10.2307/1913475 Search in Google Scholar

Gershunskaya, J., and P. Lahiri. 2010. “Robust small area estimation using a mixture model.” In Proceedings of the Joint Statistical Meeting 2010, 1 July to 5 August 2010, Vancouver, British Columbia, Canada. Available at: https://ww2.amstat.org/meetings/jsm/2010/onlineprogram/AbstractDetails.cfm?abstractid=307425 (accessed November 2021). Search in Google Scholar

Gini, C. 1914. “Sulla misura della concentrazione e della variabilita‘ dei caratteri.” In Atti del Regio Istituto Veneto di Scienze Lettere ed Arti. Available at: https://www.hetweb-site.net/het/texts/gini/gini_1914.pdf. Search in Google Scholar

Graf, M., J.M. Marín, and I. Molina. 2019. “A generalized mixed model for skewed distributions applied to small area estimation.” TEST 28 (2): 565–597. DOI: https://doi.org/10.1007/s11749-018-0594-2.10.1007/s11749-018-0594-2 Search in Google Scholar

Istat Siqual. 2008. “Information on EU-SILC survey.” Available at: http://siqual.istat.it/SIQual/visualizza.do?id=5000170&refresh=true&language=IT. Search in Google Scholar

Istat. 2017. “Occupati e disoccupati.” Available at: https://www.istat.it/it/files/2017/07/CS_Occupati-e-disoccupati_giugno_2017.pdf. Search in Google Scholar

Kreutzmann, A.-K., S. Pannier, N. Rojas-Perilla, T. Schmid, M. Templ, and N. Tzavidis. 2019. “The R package emdi for estimating and mapping regionally disaggregated indicators.” Journal of Statistical Software 91 (7): 1–33. DOI: https://doi.org/10.17169/refubium-25826. Search in Google Scholar

Langel, M., and Y. Tillè. 2013. “Variance estimation of the gini index: revisiting a result several time published.” Journal of the Royal Statistical Society A 7: 521–40. DOI: https://doi.org/10.1111/j.1467-985X.2012.01048.x.10.1111/j.1467-985X.2012.01048.x Search in Google Scholar

Lombardía, M., W. González-Manteiga, and J. Prada-Sánchez 2003. “Bootstrapping the chambers-dunstan estimate of finite population distribution function.” Journal of Statistical Planning and Inference 116: 367–388. DOI: https://doi.org/10.1016/S0378-3758(02)00240-9.10.1016/S0378-3758(02)00240-9 Search in Google Scholar

Maasoumi, E. 1986. “The measurement and decomposition of multi-dimensional inequality.” Econometrica 54: 991–97. DOI: https://doi.org/10.2307/1912849.10.2307/1912849 Search in Google Scholar

Marchetti, S., N. Tzavidis, and M. Pratesi. 2012. “Non-parametric bootstrap mean squared error estimation for m-quantile estimators of small area averages, quantiles and poverty indicators.” Computational Statistics and Data Analysis 56 (10): 2889–2902. DOI: https://doi.org/10.1016/j.csda.2012.01.023.10.1016/j.csda.2012.01.023 Search in Google Scholar

Mills, J., and S. Zandvakili. 1997. “Statistical inference via bootstrapping for measures of inequality.” Journal of Applied Econometrics 12 (2): 133–50. DOI: https://doi.org/10.1002/(SICI)1099-1255(199703)12:2,133::AID-JAE433.3.0.CO;2-H. Search in Google Scholar

Molina, I., and J. Rao. 2010. “Small area estimation of poverty indicators.” Canadian Journal of Statistics 38 (3): 369–385. DOI: https://doi.org/10.1002/cjs.10051.10.1002/cjs.10051 Search in Google Scholar

R Development Core Team. 2013. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. Available at: https://www.yumpu.com/en/document/view/6853895/r-a-language-and-environment-for-statistical-computing. (accessed November 2021). Search in Google Scholar

Rao, J., and I. Molina. 2015. Small Area Estimation. Wiley Series in Survey Methodology. Wiley.10.1002/9781118735855 Search in Google Scholar

Rojas-Perilla, N., S. Pannier, T. Schmid, and N. Tzavidis. 2020. “Data-driven transformations in small area estimation.” Journal of the Royal Statistical Series A 183 (1): 121–148. DOI: https://doi.org/10.1111/rssa.12488.10.1111/rssa.12488 Search in Google Scholar

SAMPLE (Small Area Methods for Poverty and Living Conditions). Project founded by the 7th Framwork Programme of the EU. Grant SSH - CT - 2007 – 217565. Available at: http://www.sample-project.eu. Search in Google Scholar

Schmid, T., N. Tzavidis, R. Münnich, and R.L. Chambers. 2016. “Outlier robust small area estimation under spatial correlation.” Scandinavian Journal of Statistics 43 (3): 806–826. DOI: https://doi.org/10.1111/sjos.12205.10.1111/sjos.12205 Search in Google Scholar

Shapiro, S., and M. Wilk. 1965. “An analysis of variance test for normality (complete samples).” Biometrika 67: 215–216. DOI: https://doi.org/10.2307/2333709.10.2307/2333709 Search in Google Scholar

Shorrocks, A. 1980. “The class of additively decomposable inequality measures.” Econometrica 48: 613–625. DOI: https://doi.org/10.2307/1913126.10.2307/1913126 Search in Google Scholar

Sinha, S., and J. Rao. 2009. “Robust small area estimation.” The Canadian Journal of Statistics 37 (3): 381–399. DOI: https://doi.org/10.1002/cjs.10029.10.1002/cjs.10029 Search in Google Scholar

Theil, H. 1967. Economics and Information Theory. Chicago: Rand McNally and Company. Search in Google Scholar

Tzavidis, N., S. Marchetti, and R. Chambers. 2010. “Robust estimation of small area means and quantiles.” Australian and New Zeland Journal of Statistics 52 (2): 167–186. DOI: https://doi.org/10.1111/j.1467-842X.2010.00572.x.10.1111/j.1467-842X.2010.00572.x Search in Google Scholar

Tzavidis, N., L.-C. Zhang, A. Luna, T. Schmid, and N. Rojas-Perilla. 2018. “From start to finish: a framework for the production of small area official statistics.” Journal of the Royal Statistical Society Series A 181 (4): 927–979. DOI: https://doi.org/10.1111/rssa.12364.10.1111/rssa.12364 Search in Google Scholar

Wu, C., and R. Sitter. 2001. “Variance estimator for the finite population distribution function with complete auxiliary information.” The Canadian Journal of Statistics 29. DOI: https://doi.org/10.2307/3316078.10.2307/3316078 Search in Google Scholar

Zenga, M. 2007. “Inequality curve and inequality index based on the ratios between lower and upper arithmetic means.” Statistica e Applicazioni 4: 3–27. DOI: https://doi.org/10.1400/209575. Search in Google Scholar

Recommended articles from Trend MD

Plan your remote conference with Sciendo