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A new class of change point test statistics is proposed that utilizes a weighting and trimming scheme for the cumulative sum (CUSUM) process inspired by Rényi. A thorough asymptotic analysis and simulations both demonstrate that this new class of statistics possess superior power compared to traditional change point statistics based on the CUSUM process when the change point is near the beginning or end of the sample. Generalizations of these "Rényi" statistics are also developed to test fordoi:10.6084/m9.figshare.7958891.v2 fatcat:dxxh432jqzaxzngmavhqpz2ake