Motivated by problems of frontier estimation in productivity analysis, and boundary estimation in scatter-point image analysis, we consider polynomial-based estimators of the edge of a distribution. Our aim is to develop methods for correcting polynomial-type estimators of bias, and for constructing simultaneous confidence bands for the data edge. We tackle this problem by first deriving large-sample approximations to distributions of polynomial-based edge estimators, and then developing algorithms for simulating from them so as to produce Monte Carlo approximations to the distribution of the difference between the true edge and its estimator. This involves applying representations for joint extreme value distributions. The majority of attention is focused on the parametric case, but nonparametric problems, where polynomial approximations are fitted locally, are also considered.