Modeling with the Power Variance form of the Gamma Distribution
Katherine E. Irimata *
Division of Research and Methodology, National Center for Health Statistics, Centers for Disease Control and Prevention, Hyattsville, MD, USA.
N. David Yanez
School of Public Health, Oregon Health & Science University, Portland, Oregon, USA.
Ibrahim A. Aljasser
Department of Quantitative Analysis, King Saud University, Riyadh, Saudi Arabia.
Jeffrey R. Wilson
Department of Economics, Arizona State University, Tempe, Arizona, USA.
*Author to whom correspondence should be addressed.
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.
Keywords: Positive random variables, three-parameter gamma, mean-variance relationship, Tweedie distribution