In this paper, an extension of the class to log-scale distributions that has a third shape parameter is proposed. The extended class consists of pairs of random variables related to a logarithmic or exponential transformation. A method for calculating maximum likelihood estimates of the three parameters in each distribution is proposed. The method is applied to the pair of generalized gamma distributions and generalized log-gamma, both with three parameters. The shape parameter shows that the maximum likelihood estimator is asymptotically unbiased and consistent in the mean square error.

generalized log-gamma distribution, shape parameter, location-scale distribution, maximum likelihood estimator.