In this paper, we study the problem of estimation and prediction for a two-parameter distribution with a bathtub-shaped or increasing failure rate function based on lower records and inter-record times, and based on lower records without considering the inter-record times. The maximum likelihood and Bayesian approaches are employed to estimate the unknown parameters. As it seems that the Bayes estimates cannot be derived in a closed form, the Metropolis-Hastings within Gibbs algorithm is implemented to obtain the approximate Bayes point estimates. Bayesian prediction of a future record value is also discussed. A simulation study is conducted to evaluate the proposed point and interval estimators. A real data set consisting of COVID-19 data from Iran is analyzed to illustrate the application of the theoretical results of the paper. Moreover, a simulated data example is presented. Several concluding remarks end the paper.
