Recently, a shift-independent information measure known as generalized cumulative entropy of order n (GCE n ) was proposed by Kayal (2016). In this communication, we propose a shift-dependent version of GCE n . Various properties including the effect of transformations, bounds etc. have been discussed. Several relationships of the shift-dependent GCE n with some well-known reliability measures are studied. Few characterization results are obtained. We derive an estimator for the proposed measure via empirical distribution function approach. Large sample properties of the estimator are studied when independent observations are taken from a Weibull distribution. © 2018, © 2018 Taylor & Francis Group, LLC.