Small area estimation of additive parameters under unit-level generalized linear mixed models

Average incomes and poverty proportions are additive parameters obtained as averages of a given function of an income variable. As the variable income has an asymmetric distribution, it is not properly modelled via normal distributions. When dealing with this type of variable, a first option is to apply transformations that approximate normality. A second option is to use nonsymmetric distributions from the exponential family. This paper proposes unit-level generalized linear mixed models for modelling asymmetric positive variables and for deriving three types of predictors of small area additive parameters, called empirical best, marginal and plug-in. The parameters of the introduced model are estimated by applying the maximum likelihood method to the Laplace approximation of the likelihood. The mean squared errors of the predictors are estimated by parametric bootstrap. The introduced methodology is applied and illustrated under unit-level gamma mixed models. Some simulation experiments are carried out to study the behaviour of the fitting algorithm, the small area predictors and the bootstrap estimator of the mean squared errors. By using data of the Spanish living condition survey of 2013, an application to the estimation of average incomes and poverty proportions in counties of the region of Valencia is given