Hierarchical Bayesian estimation of inequality measures with nonrectangular censored survey data with an application to wealth distribution of French households. (English) Zbl 1223.62170

Gautier, Eric

Ann. Appl. Stat. 5, No. 2B, 1632-1656 (2011).

Summary: We consider the estimation of wealth inequality measures with their confidence interval, based on survey data with interval censoring. We rely on a Bayesian hierarchical model. It consists of a model where, due to survey sampling and unit nonresponse, the summaries of the wealth distribution of households are observed with error; a mixture of multivariate models for the wealth components where groups correspond to portfolios of assets; and a prior on the parameters. A Gibbs sampler is used for numerical purposes to do the inference. We apply this strategy to the French 2004 Wealth Survey. In order to alleviate the nonresponse, the amounts were systematically collected in the form of brackets. Matched administrative data on the liability of the respondents for wealth tax and response to overview questions are used to better localize the wealth components. It implies nonrectangular multidimensional censoring. The variance of the error term in the model for the population inequality measures is obtained using linearization and taking into account the complex sampling design and the various weight adjustments.

MSC:

62P20	Applications of statistics to economics
91B82	Statistical methods; economic indices and measures
62F15	Bayesian inference
62D05	Sampling theory, sample surveys
65C60	Computational problems in statistics (MSC2010)

Keywords:

inequality; wealth distribution; survey methodology; Bayesian statistics; MCMC

References:

[1]	Atkinson, A. B. (1970). On the measurement of inequality. J. Econom. Theory 2 244-263. · doi:10.1016/0022-0531(70)90039-6
[2]	Caron, N., Deville, J. C. and Sautory, O. (1998). Estimation de précision de données issues d’enquêtes: document méthodologique sur le logiciel POULPE. Document de travail INSEE M9806.
[3]	Chand, H. and Gan, L. (2003). The effect of bracketing in wealth estimation. Review of Income and Wealth 49 273-287.
[4]	Cordier, M. and Girardot, P. (2007). Comparaison et recalage des montants de l’enquête patrimoine sur la comptabilité nationale. Document de travail INSEE F0702.
[5]	Dell, F., d’Haultfœuille, X., Février, P. and Massé, E. (2002). Mise en œuvre de calcul de variance par linéarisation. Actes des VIII emes Journées de Méthodologie Statistique . Available at .
[6]	Deville, J. C. and Särndal, C. E. (1992). Calibration estimators in survey sampling. J. Amer. Statist. Assoc. 87 376-382. · Zbl 0760.62010 · doi:10.2307/2290268
[7]	Deville, J. C. (1999). Variance estimation for complex statistics and estimators: Linearization and residual techniques. Survey Methodol. 25 193-203.
[8]	Foster, J. E. (1983). An axiomatic characterization of the Theil measure of income inequality. J. Econom. Theory 2 244-263. · Zbl 0522.90020 · doi:10.1016/0022-0531(83)90023-6
[9]	Gautier, E. (2005). Eléments sur les mécanismes de sélection dans les enquêtes et sur la non-réponse non-ignorable. Actes des IX emes Journées de Méthodologie Statistique . Available at .
[10]	Heeringa, S. G., Little, R. J. A. and Raghunathan, T. E. (2002). Multivariate imputation of coarsened survey data on household wealth. In Survey Nonresponse (R. M. Groves et al., eds.). Wiley, Hoboken, NJ.
[11]	Juster, T. F. and Smith, J. P. (1997). Improving the quality of economic data: Lessons from the HRS and AHEAD. J. Amer. Statist. Assoc. 92 1268-1278.
[12]	Kennickell, A. B. (1998). Multiple imputation in the survey of consumer finances. In Proceedings of the Section on Business and Economic Statistics, 1998 Annual Meetings of the American Statistical Association , Dallas, TX. Available at .
[13]	Little, R. J. A. and Rubin, D. B. (2002). Statistical Analysis with Missing Data , 2nd ed. Wiley, Hoboken, NJ. · Zbl 1011.62004
[14]	Matei, A. and Tillé, Y. (2005). Evaluation of variance approximations and estimators in maximum entropy sampling with unequal probability and fixed sample size. Journal of Official Statistics 21 543-570.
[15]	McCulloch, R. and Rossi, P. E. (1994). An exact likelihood analysis of the multinomial Probit model. J. Econometrics 64 207-240. · doi:10.1016/0304-4076(94)90064-7
[16]	Robert, C. (1995). Simulation of truncated normal variables. Statist. Comput. 5 121-125.
[17]	Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods , 2nd ed. Springer, New York. · Zbl 1096.62003
[18]	Roberts, G. O. and Polson, N. G. (1994). On the geometric convergence of the Gibbs sampler. J. Roy. Statist. Soc. Ser. B 56 377-384. · Zbl 0796.62029
[19]	Särndal, C. E., Swensson, B. and Wretman, J. H. (1992). Model Assisted Survey Sampling . Springer, New York. · Zbl 0742.62008
[20]	Schafer, J. L. (2001). Analysis of Incomplete Multivariate Data , 2nd ed. Chapman and Hall, London. · Zbl 0997.62510
[21]	Shao, J. (1994). L -statistics in complex survey problems. Ann. Statist. 22 946-967. · Zbl 0807.62007 · doi:10.1214/aos/1176325505
[22]	Theil, H. (1967). Economics and Information Theory . North-Holland, Amsterdam.