We have discussed elsewhere the use of optimization to minimize a goodness-of-fit statistic to determine the parameters that best fit a distribution to a data set. The most powerful use of optimization would, however, be to maximize a likelihood function because of the desirable properties a likelihood function has for parameter estimation. We offer four examples here that demonstrate the power of MLE optimization methods where we have complete or censored data.

*Examples*

We provide four examples of using MLE optimization to fit a Weibull(0,b,a) distribution to the four censored data possibilities, and with the same data set (censored to fit the circumstances) for comparison. For illustrative and comparison purposes the data were generated from a Weibull(0,6,1.5) distribution. The log likelihood function is maximized in place of the likelihood function because it produces fewer problems: Excel, like any mathematics package, has a limited number of digits at its disposal to define values, and a joint probability calculation quickly produces exceedingly small numbers. Excel array functions are used to calculate the likelihood functions because it reduces the calculation location to one cell, and you can then use that formula to obtain the joint and marginal distributions of the two parameters by creating a table.