Multivariate mixture model for small area estimation of poverty indicators

When disaggregation of national estimates in several domains or areas is required, direct survey estimators, which use only the domain‐specific survey data, are usually design‐unbiased even under complex survey designs (at least approximately) and require no model assumptions. Nevertheless, they are appropriate only for domains or areas with sufficiently large sample size. For example, when estimating poverty in a domain with a small sample size (small area), the volatility of a direct estimator might make that area seems like very poor in one period and very rich in the next one. Small area (or indirect) estimators have been developed in order to avoid such undesired instability. Small area estimators borrow strength from the other areas so as to improve the precision and therefore obtain much more stable estimators. However, the usual model‐based assumptions, which include some kind of area homogeneity, may not hold in real applications. A more flexible model based on multivariate mixtures of normal distributions that generalises the usual nested error linear regression model is proposed for estimation of general parameters in small areas. This flexibility makes the model adaptable to more general situations, where there may be areas with a different behaviour from the other ones, making the model less restrictive (hence, more close to nonparametric) and more robust to outlying areas. An expectation‐maximisation (E‐M) method is designed for fitting the proposed mixture model. Under the proposed mixture model, two different new predictors of general small area indicators are proposed. A parametric bootstrap method is used to estimate the mean squared errors of the proposed predictors. Small sample properties of the new predictors and of the bootstrap procedure are analysed by simulation studies and the new methodology is illustrated with an application to poverty mapping in Palestine.