Fitting the Generalized Beta Distribution to Data

SYNOPTIC ABSTRACT

The standard beta distribution is one of the few well-studied distributions with [0, 1] support. Oftentimes the flexibility of the standard beta is desired as a model but the [0, 1] support represents an unreasonable restriction. Accordingly, the model is transformed by location-scale and/or ratio transformations to expand the support of the distribution and/or add flexibility. The several resulting classes of distributions are referred to in an umbrella fashion as the generalized beta distributions, the parent distribution of the standard beta. The generalized beta is equivalent to the Pearson Type I distribution. The extreme flexibility of this class makes it very useful in fitting distributions to data sets similar to the way generalized lambda distributions are used. In this paper, we discuss properties and draw connections between the various forms of the generalized beta distribution. Estimators such as maximum likelihood, method of moments, and others are developed, compared, and applied to real and simulated data.

Key Words and Phrases:

• maximum likelihood estimators
• method of moments
• order statistics
• digamma and trigamma functions
• skewness
• kurtosis