Confidence Intervals of Gini Coefficient Under Unequal Probability Sampling

Yves G. Berger; İklim Gedik Balay

doi:10.2478/jos-2020-0013

Abstract

We propose an estimator for the Gini coefficient, based on a ratio of means. We show how bootstrap and empirical likelihood can be combined to construct confidence intervals. Our simulation study shows the estimator proposed is usually less biased than customary estimators. The observed coverages of the empirical likelihood confidence interval proposed are also closer to the nominal value.

References

Beach, C.M. and R. Davidson. 1983. “Distribution-free statistical inference with Lorenz curves and income shares.” The Review of Economic Studies 50(4): 723–735. DOI: https://doi.org/10.2307/2297772.10.2307/2297772
Search in Google Scholar Back to article
Berger, Y.G. 2008. “A note on the asymptotic equivalence of jackknife and linearization variance estimation for the Gini coefficient.” Journal of Official Statistics 24-4: 541 – 555. Available at: https://www.scb.se/contentassets/ca21efb41fee47d293bbee5bf7be7fb3/a-note-on-the-asymptotic-equivalence-of-jackknife-and-linearization-variance-estimation-for-the-gini-coefficient.pdf (accessed April 2020).
Search in Google Scholar Back to article
Berger, Y.G. 2018a. “An empirical likelihood approach under cluster sampling with missing observations.” Annals of the Institute of Statistical Mathematics. DOI: https://doi.org/10.1007/s10463-018-0681-x.10.1007/s10463-018-0681-x
Search in Google Scholar Back to article
Berger, Y.G. 2018b. “Empirical likelihood approaches in survey sampling.” The Survey Statistician 78: 22–31. Available at: http://isi-iass.org/home/wp-content/uploads/N78-2018-07-ISSN.pdf (accessed April 2020).
Search in Google Scholar Back to article
Berger, Y.G. and O.D.L.R. Torres. 2012. “A unified theory of empirical likelihood ratio confidence intervals for survey data with unequal probabilities.” Proceedings of the Survey Research Method Section of the American Statistical Association, Joint Statistical Meetings, San Diego: 15. Available at: http://www.asasrms.org/Proceedings (accessed November 2019).
Search in Google Scholar Back to article
Berger, Y.G. and O.D.L.R. Torres. 2014. “Empirical likelihood confidence intervals: an application to the EU-SILC household surveys.” In Contribution to Sampling Statistics, Contribution to Statistics, edited by F. Mecatti, P.L. Conti, and M.G. Ranalli, 65–84. Springer.10.1007/978-3-319-05320-2_5
Search in Google Scholar Back to article
Berger, Y.G. and O.D.L.R. Torres. 2016. “An empirical likelihood approach for inference under complex sampling design.” Journal of the Royal Statistical Society Series B 78(2): 319–341. DOI: https://doi.org/10.1111/rssb.12115.10.1111/rssb.12115
Search in Google Scholar Back to article
Bhattacharya, D. 2007. “Inference on inequality from household survey data.” Journal of Econometrics 137(2): 674–707. DOI: https://doi.org/10.1016/j.jeconom.2005.09.003.10.1016/j.jeconom.2005.09.003
Search in Google Scholar Back to article
Binder, D.A. and M.S. Kovaćević. 1995. “Estimating some measure of income inequality from survey data: an application of the estimating equation approach.” Survey Methodology 21(2): 137–145. Available at: https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X199500214396 (accessed April 2020).
Search in Google Scholar Back to article
Bishop, J., J.P. Formby, and B. Zheng. 1997. “Statistical inference and the Sen index of poverty.” International Economic Review 381–387. DOI: https://doi.org/10.2307/2527379.10.2307/2527379
Search in Google Scholar Back to article
Brent, R.P. 1973. Algorithms for Minimization without Derivatives. New-Jersey: Prentice-Hall. ISBN 0-13-022335-2.
Search in Google Scholar Back to article
Davidson, R. 2009. “Reliable inference for the Gini index.” Journal of Econometrics 150(1): 30–40. DOI: https://doi.org/10.1016/j.jeconom.2008.11.004.10.1016/j.jeconom.2008.11.004
Search in Google Scholar Back to article
Dekker, T.J. 1969. “Finding a zero by means of successive linear interpolation.” In Constructive Aspects of the Fundamental Theorem of Algebra, edited by B. Dejon and P. Henrici, 37–489. Handbook of Statistics, London: Wiley-Interscience.
Search in Google Scholar Back to article
Gajdos, T. and J.A. Weymark. 2005. “Multidimensional generalized Gini indices.” Economic Theory 26(3): 471–496. DOI: https://doi.org/10.1007/s00199-004-0529-x.10.1007/s00199-004-0529-x
Search in Google Scholar Back to article
Gastwirth, J.L. and M. Gail. 1985. “Simple asymptotically distribution-free methods for comparing Lorenz curves and Gini indices obtained from complete data.” Advances in Econometrics 4: 229–243. Available at: https://scholar.google.com/scholar_lookup?title=Simple%20Asymptotically%20Distribution-free%20Methods%20for%20Comparing%20Lorenz%20Curves%20and%20Gini%20Indices%20Obtained%20from%20Complete%20Data&author=JL.%20Gastwirth&author=MH.%20Gail&pages=229-243&publication_year=1985 (accessed April 2020).
Search in Google Scholar Back to article
Giles, D. 2004. “Calculating a standard error for the Gini coefficient: some further results.” Oxford Bulletin of Economics and Statistics 66(3): 425–433. DOI: https://doi.org/10.1111/j.1468-0084.2004.00086.x.10.1111/j.1468-0084.2004.00086.x
Search in Google Scholar Back to article
Gini, C. 1914. “Sulla Misura Della Concentrazione e Cella Variabilitá dei Caratteri.” Atti del Reale Istituto veneto di scienze, lettere ed arti, 73: 1203–1248. Available at: https://EconPapers.repec.org/RePEc:mtn:ancoec:0501 (accessed April 2020).
Search in Google Scholar Back to article
Giorgi, G.M. and C. Gigliarano. 2017. “The Gini concentration index: a review of the inference literature.” Journal of Economic Surveys 31(4): 1130–1148. DOI: https://doi.org/10.1111/joes.12185.10.1111/joes.12185
Search in Google Scholar Back to article
Glasser, G.J. 1962. “Variance formulas for the mean difference and coefficient of concentration.” Journal of the American Statistical Association 57(299): 648–654. DOI: https://doi.org/10.1080/01621459.1962.10500553.10.1080/01621459.1962.10500553
Search in Google Scholar Back to article
Hájek, J. 1971. “Comment on a paper by Basu, D.” In Foundations of Statistical Inference, edited by V.P. Godambe and D.A. Sprott, 236. Toronto: Holt, Rinehart & Winston.10.2307/2334645
Search in Google Scholar Back to article
Hartley, H.O. and J.N.K. Rao. 1962. “Sampling with unequal probabilities without replacement.” The Annals of Mathematical Statistics 33: 350–374. DOI: https://doi.org/10.1214/aoms/1177704564.10.1214/aoms/1177704564
Search in Google Scholar Back to article
Hoeffding, W. 1948. “A non-parametric test of independence.” The annals of Mathematical Statistics, 546–557. DOI: https://doi.org/10.1214/aoms/1177730150.10.1214/aoms/1177730150
Search in Google Scholar Back to article
Karagiannis, E. and M. Kovaćević. 2000. “A method to calculate the Jack-250 knife variance estimator for the Gini coefficient.” Oxford Bulletin of Economics and Statistics 62(1): 119–122. DOI: https://doi.org/10.1111/1468-0084.00163.10.1111/1468-0084.00163
Search in Google Scholar Back to article
Koshevoy, G. and K. Mosler. 1997. “Multivariate Gini indices.” Journal of Multivariate Analysis 60(2): 252–276. DOI: https://doi.org/10.1006/jmva.1996.1655.10.1006/jmva.1996.1655
Search in Google Scholar Back to article
Langel, M. and Y. Tillé. 2013. “Variance estimation of the Gini index: revisiting a result several times published.” Journal of the Royal Statistical Society: Series A (Statistics in Society) 176(2): 521–540. DOI: https://doi.org/10.1111/j.1467-985X.2012.01048.x.10.1111/j.1467-985X.2012.01048.x
Search in Google Scholar Back to article
Lorenz, M.O. 1905. “Methods of measuring the concentration of wealth.” Publications of the American Statistical Association 9(70): 209–219. DOI: https://doi.org/10.2307/2276207.10.2307/2276207
Search in Google Scholar Back to article
Mills, J.A. and S. Zandvakili. 1997. “Statistical inference via bootstrapping for measures of inequality.” Journal of Applied Econometrics 12(2): 133 – 150. DOI: https://doi.org/10.1002/(SICI)1099-1255(199703)12:2133:AID-JAE4333.0.CO;2-H.
Search in Google Scholar Back to article
Modarres, R. and J.L. Gastwirth. 2006. “A cautionary note on estimating the standard error of the Gini index of inequality.” Oxford Bulletin of Economics and Statistics 68(3): 385–390. DOI: https://doi.org/10.1111/j.1468-0084.2006.00167.x.10.1111/j.1468-0084.2006.00167.x
Search in Google Scholar Back to article
Nair, U.S. 1936. “The standard error of Gini’s mean difference.” Biometrika 28(3/4): 428–436. DOI: https://doi.org/10.1093/biomet/28.3-4.428.10.1093/biomet/28.3-4.428
Search in Google Scholar Back to article
Neyman, J. 1938. “On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection.” Journal of the Royal Statistical Society 97(4): 558–625. DOI: https://doi.org/10.2307/2342192.10.2307/2342192
Search in Google Scholar Back to article
Nygård, F. and A. Sandström. 1989. “Income inequality measures based on sample surveys.” Journal of Econometrics 42(1): 81–95. DOI: https://doi.org/10.1016/0304-4076(89)90077-8.10.1016/0304-4076(89)90077-8
Search in Google Scholar Back to article
Ogwang, T. 2000. “A convenient method of computing the Gini index and its standard error.” Oxford Bulletin of Economics and Statistics 62(1): 123 – 129. DOI: https://doi.org/10.1111/1468-0084.00164.10.1111/1468-0084.00164
Search in Google Scholar Back to article
Owen, A.B. 1988. “Empirical Likelihood Ratio Confidence Intervals for a Single Functional.” Biometrika 75(2): 237 – 249. DOI: https://doi.org/10.1093/biomet/75.2.237.10.1093/biomet/75.2.237
Search in Google Scholar Back to article
Peng, L. 2011. “Empirical likelihood methods for the Gini index.” Australian & New Zealand Journal of Statistics 53(2): 131–139. DOI: https://doi.org/10.1111/j.1467-842X.2011.00614.x.10.1111/j.1467-842X.2011.00614.x
Search in Google Scholar Back to article
Qin, Y., J.N.K. Rao, and C. Wu. 2010. “Empirical likelihood confidence intervals for the Gini measure of income inequality.” Economic Modelling 27: 1429–1435. DOI: https://doi.org/10.1016/j.econmod.2010.07.015.10.1016/j.econmod.2010.07.015
Search in Google Scholar Back to article
Rao, J.N.K., C.F.J. Wu, and K. Yue. 1992. “Some recent work on resampling methods for complex surveys.” Survey Methodology 18: 209–217. Available at: https://www150.statcan.gc.ca/n1/en/catalogue/12-001-X199200214486 (accessed April 2020).
Search in Google Scholar Back to article
Rust, K.F. and J.N.K. Rao. 1996. “Variance estimation for complex surveys using replication techniques.” Biometrika 5(3): 281310. DOI: https://doi.org/10.1177/096228029600500305.10.1177/0962280296005003058931197
Search in Google Scholar Back to article
Sandström, A., B. Waldén, and J.H. Wretman. 1985. Variance estimators of the Gini coefficient: simple random sampling. Sweden. Available at: https://www.scb.se/contentassets/7004f1a7effe4e8690548165695b1864/pm-p-1985-17-green.pdf (accessed April 2020).
Search in Google Scholar Back to article
Sandström, A., J.H. Wretman, and B. Waldén. 1988. “Variance estimators of the Gini coefficient – probability sampling.” Journal of Business & Economic Statistics 6(1): 113–119. DOI: https://doi.org/10.1080/07350015.1988.10509643.10.1080/07350015.1988.10509643
Search in Google Scholar Back to article
Schezhtman, E. and S. Yitzhaki. 1987. “A Measure of Association Based on Gini’s Mean Difference.” Communications in Statistics-Theory and Methods 16(1): 207–231. DOI: https://doi.org/10.1080/03610928708829359.10.1080/03610928708829359
Search in Google Scholar Back to article
Sendler, W. 1979. “On statistical inference in concentration measurement.” Metrika 26(1): 109–122. DOI: https://doi.org/10.1007/BF01893478.10.1007/BF01893478
Search in Google Scholar Back to article
Shao, J. 1994. “L-Statistics in Complex Survey Problems.” The Annals of Statistics 22(2): 946–967. DOI: https://doi.org/10.1214/aos/1176325505.10.1214/aos/1176325505
Search in Google Scholar Back to article
Wu, C. and J.N.K. Rao. 2006. “Pseudo-empirical likelihood ratio confidence intervals for complex surveys.” Canadian Journal of Statistics 34(3): 359–375. DOI: https://doi.org/10.1002/cjs.5550340301.10.1002/cjs.5550340301
Search in Google Scholar Back to article
Yitzhaki, S. 1991. “Calculating jackknife variance estimators for parameters of the Gini method.” Journal of Business & Economic Statistics 9(2): 235 – 239. DOI: https://doi.org/10.1080/07350015.1991.10509849.10.1080/07350015.1991.10509849
Search in Google Scholar Back to article
Yitzhaki, S. 1998. “More than a dozen alternative ways of spelling Gini.” Research on Economic Inequality 8: 13–30. DOI: https://doi.org/10.1.1.365.4196.
Search in Google Scholar Back to article

Confidence Intervals of Gini Coefficient Under Unequal Probability Sampling

Abstract

Paradigm

My account