We consider inference upon unknown parameters of the family of inverted exponentiated distributions when it is known that data are doubly censored. Maximum likelihood and Bayes estimates under different loss functions are derived for estimating the parameters. We use Metropolis-Hastings algorithm to draw Markov chain Monte Carlo samples, which are used to compute the Bayes estimates and construct the Bayesian credible intervals. Further, we present point and interval predictions of the censored data using the Bayesian approach. The performance of proposed methods of estimation and prediction are investigated using simulation studies, and two illustrative examples are discussed in support of the suggested methods. Finally, we propose the optimal plans under double censoring scheme.