Although skew distributions are common in for example business surveys, estimators that take this skewness into account in an explicit way are rarely used. Instead, methods that aim at lowering the mean squared error by dampening the effect of large sample “outliers” are common. These methods will by construction result in estimators with a negative bias. The issue explored in this presentation is whether modeling only the right tail of the population can yield the same, or a larger, decrease in mean squared error but with a smaller systematic bias component. I will describe a couple of estimators of this type that has been suggested in the literature, as well as one that has not been proposed before, and show preliminary results from a simulation study. Focus is on the most basic scenario where no auxiliary data is assumed to be available for design or estimation purposes.