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.

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The precise investigation of the Moore and Bilikam (MB) family of distributions is essential since they cover several distributions as special cases. We evaluate different estimation procedures for parameters of MB family comprehensively. Moreover, the confidence interval and highest posterior density interval estimators of parameters are provided. The different estimation methods of the parameters of two certain cases of the MB family are compared via the simulation data. Based on the simulation approach, we realized the estimates are convergent to the real values of the parameters and the root mean squared error (RMSE) values derived by the expectation-maximization (EM) algorithm are less than Newton–Raphson (NR), stochastic EM (SEM) and Bayes estimators. Also, the higher values of the sample size lead to better estimates in the sense close to true parameter values and having smaller RMSE values. The reliability analysis of MB family is another topic in this paper, with simulation discussion. The Monte Carlo simulations are provided to assess the performances of estimation methods. We concluded the Bayesian shrinkage estimators are better than other methods of the reliability function, which have the smallest values of the RMSE. The analysis of real dataset has been presented for illustrative purposes, which confirmed simulation results.