In this paper, exponentiated-G family of distributions is constructed as a new family of discrete distribution, using the general approach of discretization of a continuous distribution. Some statistical properties including quantiles, mean residual life, mean time to failure, mean time between failure, availability, Re ́nyi entropy, moments and order statistics are obtained. Discrete exponentiated inverted Topp–Leone distribution as a member of this family is studied in detail. Maximum likelihood approach is applied under Type-II censored sample for estimating the unknown parameters of the exponentiated inverted Topp–Leone distribution. Also, maximum likelihood estimators of the survival, hazard rate and alternative hazard rate functions are derived. The confidence intervals for the parameters, survival, hazard rate and alternative hazard rate functions are obtained. A simulation study is performed to investigate the accuracy of the theoretical results. Finally, two real data sets are analyzed to illustrate the flexibility and applicability of the proposed model for real-life applications.