Abstract
In this paper, semiparametric monotone mixed models are introduced, exploring, in particular, the problems of estimation and bootstrapping. The models are defined in a small area setting, using the assumption that some of the auxiliary variables have a monotone relationship with the response, and with the incorporation of linear terms to model other auxiliaries as the dummy variables. An estimator for the variance of the random effects is proposed and two bootstrap approaches, specially designed for monotone regression, are given to estimate the mean squared error for the area means. A simulation experiment is carried out to compare the performance of the new model-based estimators against the Fay–Herriot approach and to confirm the good performance of the bootstrap. The semiparametric model-based area estimators are also compared with the parametric-based estimators using data on a survey of lakes, where the questions of the prediction of missing data and model selection are nicely solved using simple proposals.