The Moore and Bilikam family includes lifetime distributions, hence there is a need for a meticulous investigation of the proposed family. We evaluate different estimation procedures for both parameters and reliability function of the Moore and Bilikam family comprehensively, including the maximum likelihood, Bayesian and E-Bayesian estimation methods. The estimation methods of the Moore and Bilikam family are compared via the simulation data, whereas simulation results of the Burr XII sub-model are reported. Based on the simulation approach, we concluded the estimates of the Moore and Bilikam family are convergent to the corresponding parameters, and the root mean square error values derived by the E-Bayesian method are less than other estimators. The analysis of the time between failures of secondary reactor pumps data set has been represented for illustrative purposes, which confirmed simulation results.