The quality of estimation of tail parameters, such as tail index in the univariate case, or the spectral measure in the multivariate case,
depends crucially on the part of the sample included in the estimation. A simple approach involving sequential statistical testing is proposed in order to choose this part of the sample. This method can be used both in the univariate and multivariate cases. It is computationally efficient, and can be easily automated. No visual inspection of the data is required. We establish consistency of the Hill estimator when used in conjunction with the proposed method, as well describe its asymptotic fluctuations. We compare our method to existing methods in univariate and multivariate tail estimation, and use it to analyze Danish fire insurance data.