  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 consisting of bracketed
and sometimes missing data, over a subsample of the population of interest. We
present in this article the statistical methodology 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 inequality 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.
