Transformation and Smoothing in Sample Survey Data

We consider model-based prediction of a finite population total when a monotone transformation of the survey variable makes it appropriate to assume additive, homoscedastic errors. As the transformation to achieve this does not necessarily simultaneously produce an easily parameterized mean function, we assume only that the mean is a smooth function of the auxiliary variable and estimate it non-parametrically. The back transformation of predictions obtained on the transformed scale introduces bias which we remove using smearing. We obtain an asymptotic expansion for the prediction error which shows that prediction bias is asymptotically negligible and the prediction mean-squared error (MSE) using a non-parametric model remains in the same order as when a parametric model is adopted. The expansion also shows the effect of smearing on the prediction MSE and can be used to compute the asymptotic prediction MSE. We propose a model-based bootstrap estimate of the prediction MSE. The predictor produces competitive results in terms of bias and prediction MSE in a simulation study, and performs well on a population constructed from an Australian farm survey.