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Bayesian Estimation of Inequalities with Non-Rectangular Censored Survey Data.pdf

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Bayesian Estimation of Inequalities with Non-Rectangular Censored Survey Data

a r X i v : 0 8 0 2 .4 1 9 0 v 1 [ s t a t .A P ] 2 8 F e b 2 0 0 8 Bayesian Estimation of Inequalities with Non-Rectangular Censored Survey Data Eric Gautier ENSAE - CREST, 3 avenue Pierre Larousse, 92245 Malakoff Cedex, France; gautier@ensae.fr Abstract Synthetic indices are used in Economics to measure various aspects of monetary inequalities. These scalar indices take as input the distribution over a finite population, for example the population of a specific country. In this article we consider the case of the French 2004 Wealth survey. We have at hand a partial measurement on the distribution of interest con- sisting of bracketed and sometimes missing data, over a subsample of the population of interest. We present in this article the statistical method- ology used to obtain point and interval estimates taking into account the various uncertainties. The inequality indices being nonlinear in the input distribution, we rely on a simulation based approach where the model for the wealth per household is multivariate. Using the survey data as well as matched auxiliary tax declarations data, we have at hand a quite intricate non-rectangle multidimensional censoring. For practical issues we use a Bayesian approach. Inference using Monte-Carlo approximations relies on a Monte-Carlo Markov chain algorithm namely the Gibbs sampler. The quantities interesting to the decision maker are taken to be the various in- equality indices for the French population. Their distribution conditional on the data of the subsample are assumed to be normal centered on the design-based estimates with variance computed through linearization and taking into account the sample design and total nonresponse. Exogeneous selection of the subsample, in particular the nonresponse mechanism, is assumed and we condition on the adequate covariates. Key Words: Inequality; Wealth distribution; Survey methodology; Bayesian statistics; Monte-Carlo Markov chains. AMS 2000 Subject Classification: 62