Small area estimation for business surveys: a comparison of transformation-based unit level models

Small area estimation methods are generally based on models which have assumptions of normal errors, but many types of data do not follow a normal distribution. Several approaches have been suggested to deal with skewed data, including transformations (with and without bias correction), robust models which are less affected by the tails of the distributions and building models directly with skewed error distributions. We investigate the properties of models for transformed data with a real data set which mimics a structural business survey. This contributes to the understanding of which tools are best for small area estimation with skewed data. We also investigate the sensitivity of results to different shift parameters (commonly used to make methods practical when data contain zeroes) and transformation parameters. The empirical best predictor (EBP) approach is found to be a flexible way to fit transformation-based models without the need for development of bias adjustments in back transformation. We prefer the EBP log-shift and EBP dual power which have good performance in our example (noting that the variables affecting the weighting are included in the model) because of their adaptability to new datasets. The bias-corrected empirical best (EBbc) estimator has similar performance in our example, but is tailored to the log transformation.