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Concept
Generalized beta distribution
Summary
In probability and statistics, the generalized beta distribution is a continuous probability distribution with four shape parameters (however it's customary to make explicit the scale parameter as a fifth parameter, while the location parameter is usually left implicit), including more than thirty named distributions as limiting or special cases. It has been used in the modeling of income distribution, stock returns, as well as in regression analysis. The exponential generalized beta (EGB) distribution follows directly from the GB and generalizes other common distributions.
A generalized beta random variable, Y, is defined by the following probability density function:
and zero otherwise. Here the parameters satisfy , and , , and positive. The function B(p,q) is the beta function. The parameter is the scale parameter and can thus be set to without loss of generality, but it is usually made explicit as in the function above (while the location parameter is usually left implicit and set to as in the function above).
It can be shown that the hth moment can be expressed as follows:
where denotes the hypergeometric series (which converges for all h if c
Official source
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