Statistical inference for Birnbaum-Saunders and Weibull distributions fitted to grouped and ungrouped data

For a given sample of grouped and ungrouped (raw) data, the maximum likelihood (ML) estimator is obtained using iterative algorithms such as Newton -Raphson (NR), which may not be converged always. Three-parameter Birnbaum-Saunders (BS) and Weibull distributions are frequently used in forestry and environmental sciences. In this study, we suggest using the expectation-maximization (EM) algorithm to estimate the parameters of BS and Weibull distributions when these models are fitted to grouped data. The EM algorithm is an iterative procedure that is used to obtain the ML estimator and always converges, whereas it is shown through simulation that the NR method may fail to converge. We demonstrate through three illustrations that the EM algorithm applied to the grouped data works efficiently. For the first illustration, the ML estimates of the grouped data exist and they are almost the same as the output of the EM algorithm. In the second and third real data examples that are of small sizes, the ML estimator does not exist for the ungrouped data but, we find it using the EM algorithm applied to the grouped data.