Modeling with the Power Variance form of the Gamma Distribution

Abstract

It is not uncommon to encounter data where the distribution of the responses is not known to completely follow any of the common probability models. While there are general classes of models, such as the Tweedie distribution, which can be adopted in such cases, many approximations have been proposed based on the fact that they are often easier to obtain. We bring to the discussion a three-parameter power variance representation of the gamma distribution Γ(α, β) that has a general mean-variance relationship , where μ = E(Y) is the mean or expected value of Y,  is a scale parameter, and  is the degree of power of the expression. This power variance formulation is a flexible extension of the gamma distribution, and are used to approximate various models and determine significant predictors even when the distribution is not fully realized. We present a comparison of the power variance model to several known distributions which have similar mean-variance. In addition, we provide a more general representation of the relation , where  is the variance function indexed by the parameter . We demonstrate the performance of the power variance modeling approach through a simulation and evaluate two numerical examples, including high school absenteeism and concrete compression strength.