Corrado Gini, a pioneer in balanced sampling and inequality theory. (English) Zbl 1329.62020
Summary: This paper attempts to make the link between two of Corrado Gini’s contributions to statistics: the famous inequality measure that bears his name and his work in the early days of balanced sampling. Some important notions of the history of sampling such as representativeness, randomness, and purposive selection are clarified before balanced sampling is introduced. The Gini index is described, as well as its estimation and variance estimation in the sampling framework. Finally, theoretical grounds and some simulations on real data show how some well used auxiliary information and balanced sampling can enhance the accuracy of the estimation of the Gini index.
MSC:
| 62-03 | History of statistics |
| 91-03 | History of game theory, economics, and finance |
| 62D05 | Sampling theory, sample surveys |
| 62P20 | Applications of statistics to economics |
| 91B82 | Statistical methods; economic indices and measures |
| 01A70 | Biographies, obituaries, personalia, bibliographies |
Biographic References:
Gini, CorradoReferences:
| [1] | Berger, Y. G. (2008) A note on asymptotic equivalence of jackknife and linearization variance estimation for the Gini coefficient, Journal of Official Statistics, 24, 541–555. |
| [2] | Binder, D. A. (1996) Linearization methods for single phase and two-phase samples: a cookbook approach, Survey Methodology, 22, 17–22. |
| [3] | Binder, D. A. and Patak, Z. (1994) Use of estimating functions for estimation from complex surveys, Journal of the American Statistical Association, 89, 1035–1043. · Zbl 0825.62392 · doi:10.1080/01621459.1994.10476839 |
| [4] | Cowell, F. A. (1988) Inequality decomposition: Three bad measures, Bulletin of Economic Research, 40, 309–311. · doi:10.1111/j.1467-8586.1988.tb00274.x |
| [5] | Dalton, H. (1920) The measurement of the inequality of incomes, The Economic Journal, 30, 348–361. · doi:10.2307/2223525 |
| [6] | Dell, F., d’Haultfoeuille X., Fevrier, P. and Masse, E. (2002) Mise en oeuvre du calcul de variance par linéarisation, INSEE-Methodes: Actes des Journees de Methodologie Statistique, 73–104. |
| [7] | Demnati, A. and Rao, J. N. K. (2004) Linearization variance estimators for survey data (with discussion), Survey Methodology, 30, 17–34. |
| [8] | Deville, J.-C. (1988) Estimation linéaire et redressement sur informations auxiliaires d’enquêtes par sondage, In: Essais en l’honneur d’Edmond Malinvaud (Eds. A. Monfort, and J. J. Laffond), Economica, Paris, pp. 915–929. |
| [9] | Deville, J.-C. (1992) Constrained samples, conditional inference, weighting: Three aspects of the utilisation of auxiliary information., In: Proceedings of the Workshop on the Uses of Auxiliary Information in Surveys, Örebro, Sweden. |
| [10] | Deville, J.-C. (1996) Estimation de la variance du coefficient de Gini estimé par sondage, Actes des journées de Méthodologie Statistique, INSEE, 69-70-71, 269–288. |
| [11] | Deville, J.-C. (1999) Variance estimation for complex statistics and estimators: linearization and residual techniques, Survey Methodology, 25, 193–204. |
| [12] | Deville, J.-C, Grosbras, J-M. and Roth, N. (1988) Efficient sampling algorithms and balanced sample, In: COMPSTAT, Proceedings in Computational Statistics, Physica Verlag, Heidelberg, pp. 255–266. |
| [13] | Deville, J.-C. and Tillé, Y. (2004) Efficient balanced sampling: The cube method, Biometrika, 91, 893–912. · Zbl 1064.62015 · doi:10.1093/biomet/91.4.893 |
| [14] | Deville, J.-C. and Tillé, Y. (2005) Variance approximation under balanced sampling, Journal of Statistical Planning and Inference, 128, 569–591. · Zbl 1089.62005 · doi:10.1016/j.jspi.2003.11.011 |
| [15] | Eurostat (2005) The continuity of indicators during the transition between ECHP and EU-SILC, Technical report, Working Papers and Studies, Luxembourg: Office for Official Publications of the European Communities. |
| [16] | Fienberg, S. E. and Tanur, J. M. (1995) Reconsidering Neymanon experimentation and sampling: Controversies and fundamental contributions, Probability and Mathematical Statistics, 15, 47–60. |
| [17] | Fienberg, S. E. and Tanur, J. M. (1996) Reconsidering the fundamental contributions of Fisher and Neyman on experimentation and sampling, International Statistical Review, 64, 237–253. · Zbl 0899.62003 · doi:10.2307/1403784 |
| [18] | Fuller, W. A. (2009) Some design properties of a rejective sampling procedure, Biometrika, 96, 933–944. · Zbl 1179.62020 · doi:10.1093/biomet/asp042 |
| [19] | Galvani, L. (1951) Révision critique de certains points de la méthode représentative, Revue de l’Institut International de Statistique, 19, 1–12. · Zbl 0043.13603 · doi:10.2307/1401498 |
| [20] | Giles, D. E. A. (2004) Calculating a standard error for the Gini coefficient: some further results, Oxford Bulletin of Economics and Statistics, 66, 425–433. · doi:10.1111/j.1468-0084.2004.00086.x |
| [21] | Gini, C. (1912) Variabilità e Mutabilità, Tipografia di Paolo Cuppin, Bologna. |
| [22] | Gini, C. (1914) Sulla misura della concentrazione e della variabilità dei caratteri, Atti del R. Istituto Veneto di SS. LL. AA, 73, 1203–1248. |
| [23] | Gini, C. (1921) Measurement of inequality and incomes, The Economic Journal, 31, 124–126. · doi:10.2307/2223319 |
| [24] | Gini, C. and Galvani, L. (1929) Di un’applicazione del metodo rappresentativo all’ ultimo censimento Italiano della popolazione (1 dicembre 1921), Annali di Statistica, Serie VI, 4, 1–107. |
| [25] | Glasser, G. (1962) Variance formulas for the mean difference and coefficient of concentration, Journal of the American Statistical Association, 57, 648–654. · Zbl 0113.13803 · doi:10.1080/01621459.1962.10500553 |
| [26] | Hájek, J. (1959) Optimum strategy and other problems in probability sampling, Casopispro Pestování Matematiky, 84, 387–423. |
| [27] | Hájek, J. (1981) Sampling from a Finite Population, Marcel Dekker, New York. · Zbl 0494.62008 |
| [28] | Hampel, F. R., Ronchetti, E., Rousseuw, P. J. and Stahel, W. (1985) Robust Statistics: The Approach Based on the Influence Function, Wiley, New-York. |
| [29] | Hansen, M H. (1987) Some history and reminiscences on survey sampling, Statistical Science, 2, 180–190. · Zbl 0955.62537 · doi:10.1214/ss/1177013352 |
| [30] | Hansen, M. H., Dalenius, T. D., and Tepping, B. J. (1985) The development of sample survey in finite population, In: A Celebration of Statistics (Eds. A. Atkinson and S. Fienberg), The ISI Centenary Volume, Springer, pp. 327–353. |
| [31] | Hedayat, A. S. and Majumdar, D. (1995) Generating desirable sampling plans by the technique of trade-off in experimental design, Journal of Statistical Planning and Inference, 44, 237–247. · Zbl 0812.62009 · doi:10.1016/0378-3758(94)00044-V |
| [32] | Hoeffding, W. (1948) A class of statistics with asymptotically normal distribution, Annals of Mathematical Statistics, 19, 293–325. · Zbl 0032.04101 · doi:10.1214/aoms/1177730196 |
| [33] | Hulliger, B. and Munnich, R. (2006) Variance estimation for complex surveys in the presence of outliers, In: Proceedings of the Section on Survey Research Methods, pp. 3153–3161, American Statistical Association. |
| [34] | Hyndman, R. J. and Fan, Y. (1996) Sample quantiles in statistical packages, American Statistician, 50, 361–365. |
| [35] | Jensen, A. (1926) Report on the representative method in statistics, Bulletin of the International Statistical Institute, 22, 359–380. |
| [36] | Kiaer, A. (1896) Observations et expériences concernant des dénombrements représentatifs, Bulletin de l’Institut International de Statistique, 9, 176–183. |
| [37] | Kiaer, A. (1899) Surles méthodes représentatives ou typologiques appliquées à la statistique, Bulletin de l’Institut International de Statistique, 11, 180–185. |
| [38] | Kiaer, A. (1903) Sur les méthodes représentatives ou typologiques appliquées à la statistique, Bulletin de l’Institut International de Statistique, 31, 66–78. |
| [39] | Kiaer, A. (1905) Discours sans intitulée sur la méethode représentative, Bulletin de l’Institut International de Statistique, 14, 119–134. |
| [40] | Kovacevic, M. S. and Binder, D. A. (1997) Variance estimation for measures of income inequality and polarization – the estimating equations approach, Journal of Official Statistics, 13, 41–58. |
| [41] | Kruskal, W. and Mosteller, F. (1979a) Representative sampling, I: Nonscientific literature, International Statistical Review, 47, 13–24. · Zbl 0471.62010 · doi:10.2307/1403202 |
| [42] | Kruskal, W. and Mosteller, F. (1979b) Representative sampling, II: Scientific literature, excluding statistics, International Statistical Review, 47, 111–127. · Zbl 0471.62010 · doi:10.2307/1402564 |
| [43] | Kruskal, W. and Mosteller, F. (1979c) Representative sampling, III: The current statistical literature, International Statistical Review, 47, 245–265. · Zbl 0471.62010 · doi:10.2307/1402647 |
| [44] | Kruskal, W. and Mosteller, F. (1980) Representative sampling, IV: The history of the concept in statistics, International Statistical Review, 48, 169–195. · Zbl 0471.62011 · doi:10.2307/1403151 |
| [45] | Langel, M. and Tillé, Y. (2011) Statisticalinferenceforthequintileshareratio, Journal of Statistical Planning and Inference, 141, 2976–2985. · Zbl 1213.62188 · doi:10.1016/j.jspi.2011.03.023 |
| [46] | Lesage, E. (2008) Contraintes d’équilibrage non linéaires, In: Méthodes d’enquêtes: applications aux enquêtes longitudinales, à la santé et aux enquêtes électorales, pp. 285–289, (Eds. R. Guilbert, D. Haziza, D. A. Ruiz-Gazen, Y. Tillé), Dunod, Paris. |
| [47] | Lorenz, M. O. (1905) Methods of measuring the concentration of wealth, Publications of the American Statistical Association, 9, 209–219. · doi:10.2307/2276207 |
| [48] | Monti, A. C. (1991) The study of the Gini concentration ratio by means of the influence function, Statistica, 51, 561–577. · Zbl 0751.62053 |
| [49] | Nedyalkova, D. and Tillé, Y. (2009) Optimal sampling and estimation strategies under linear model, Biometrika, 95, 521–537. · Zbl 1437.62562 · doi:10.1093/biomet/asn027 |
| [50] | Neyman, J. (1934) On the two different aspects of representative method: The method of stratified sampling and the method of purposive selection, Journal of the Royal Statistical Society, 97, 558–606. · JFM 61.1310.02 · doi:10.2307/2342192 |
| [51] | Neyman, J. (1952) Lectures and Conferences on Mathematical Statistics and Probability, Graduate School; U. S. Department of Agriculture, Washington. · Zbl 0049.09802 |
| [52] | Nygard, F. and Sandstrom, A. (1985) The estimation of the Gini and the entropy inequality parameters in finite populations, Journal of Official Statistics, 1, 399–412. |
| [53] | Ogwang, T. (2000) A convenient method of computing the Gini index and its standard error, Oxford Bulletin of Economics and Statistics, 62, 123–129. · doi:10.1111/1468-0084.00164 |
| [54] | Osier, G. (2006) Variance estimation: the linearization approach applied by Eurostat to the 2004 SILC operation, Technical report, Eurostat and Statistics Finland Methodological Workshop on EU-SILC, Helsinki, 7–8 November 2006. |
| [55] | Osier, G. (2009) Variance estimation for complex indicators of poverty and inequality using linearization techniques, Survey Research Methods, 3, 167–195. |
| [56] | Pigou, A. C. (1912) Wealth and Welfare, McMillan, London. |
| [57] | Royall, R. M. and Herson, J. (1973) Robust estimation in finite populations I, Journal of the American Statistical Association, 68, 880–889. · Zbl 0272.62005 · doi:10.1080/01621459.1973.10481440 |
| [58] | Sandstrom, A., Wretman, J. H., and Walden, B. (1985) Variance estimators of the Gini coefficient: Simple random sampling, Metron, 43, 41–70. · Zbl 0607.62140 |
| [59] | Sandstrom, A., Wretman, J. H., and Walden, B. (1988) Variance estimators of the Gini coefficient: Probability sampling, Journal of Business and Economic Statistics, 6, 113–120. |
| [60] | Shorrocks, A. F. (1980) The class of additive decomposable inequality measures, Economica, 48, 613–625. · Zbl 0435.90040 |
| [61] | Shorrocks, A. F. (1984) Inequality decomposition by population subgroups, Econometrica, 52, 1369–1385. · Zbl 0601.90029 · doi:10.2307/1913511 |
| [62] | Theil, H. (1967) Economics and Information Theory, Rand McNally. |
| [63] | Theil, H. (1969) The desired political entropy, The American Political Science Review, 63, 521–525. · doi:10.2307/1954705 |
| [64] | Tillé, Y. and Favre, A.-C. (2004) Co-ordination, combination and extension of optimal balanced samples, Biometrika, 91, 913–927. · Zbl 1064.62019 · doi:10.1093/biomet/91.4.913 |
| [65] | Tillé, Y. and Favre, A. C. (2005) Optimal allocation in balanced sampling, Statistics and Probability Letters, 74, 31–37. · Zbl 1123.62011 · doi:10.1016/j.spl.2005.04.027 |
| [66] | Woodruff, R. S. (1971) A simple method for approximating the variance of a complicated estimate, Journal of the American Statistical Association, 66, 411–414. · Zbl 0216.47602 · doi:10.1080/01621459.1971.10482279 |
| [67] | Xu, K. (2003) How has the literature on Gini’s index evolved in the past 80 years?, Working papers archive, Dalhousie, Department of Economics. |
| [68] | Yates, F. (1960) Sampling Methods for Censuses and Surveys, Charles Griffin, London, third edition. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
