Estimation problems associated with the three parameter gamma distribution

Abstract

A previous study considered by simulation the first four moments of the maximum likelihood estimators of three parameters, threshold, shape, and scale. The distribution of scale, as far as four moments go, was not unduly nonnormal; however this was not the case for the other two parameters, so that transformations to alleviate non-normality were introduced. Reverting to characteristics of the basic estimators through moments posed problems. Here use is made of the Pearson-Tukey approximations which depend on basic percentage points (or percentiles such as 5%, 50%, 95%); these can be assessed using four-moment frequency models (Pearson, Johnson translation system, etc). However the Pearson-Tukey study does not include skewness and kurto-sis, a gap still awaiting an answer. In addition to bringing forward properties of the means and standard deviations of the three cases, we transform the densities using elementary transformations. To heighten the impact, graphics of the densities are given for a set of basic parameters, and some sample sizes. Lastly, mention is made of the perennial problem of asymptotic means and variances used for supposedly large enough sample sizes.

Keywords:

• density graphics
• density models
• four moment models
• Johnson translation system
• Pearson curves
• Pearson-Tukey percentiles, simulation