Estimation of poverty indicators in small areas under skewed distributions

The standard methods for poverty mapping at local level assume
that incomes follow a log-normal model. However, the log-normal distribution is not always well suited for modeling the income, which often shows skewness even at the log scale. As an alternative, we propose to consider a much more flexible distribution called generalized beta distribution of the second kind (GB2). The flexibility of the GB2 distribution arises from the fact that it contains four parameters in contrast with the two parameters of the log normal. One of the parameters of the GB2 controls the shape of left tail and another controls the shape of the right tail, making it suitable to model different forms of skewness. In particular, it includes the log-normal distribution as a limiting case. In this sense, it can be seen as an extension of the log-normal model to handle more adequately potential atypical or extreme values and it has been successfully applied to model the income. We propose a small area model for the incomes based on a multivariate extension of the GB2 distribution. Under this model, we define empirical best (EB) estimators of general non-linear area parameters; in particular, poverty indicators and we describe how to obtain Monte Carlo approximations of the EB estimators. A parametric bootstrap procedure is proposed for estimation of the mean squared error.