incomplete Beta function

Yule distribution is one of the more accurate distributions for fitting the heavy tailed data, although it is difficult to get a closed formula for the parameter estimator. In Spierdijk (2007), single maximum likelihood estimator (MLE) for the Yule (ρ) parameter was found numerically. In addition, another extension of the distribution was derived and it denoted as GYule (ρ,α), generalized Yule distribution. The moment estimators were got numerically for GYule but it was difficult to get the MLE's for the parameters. [Spierdijk (2007)]. In this study, single MLE’s for GYule (ρ,α) parameters was found numerically and was applied for the same dataset used in Spierdijk (2007). GYule density function was also derived using the method of mixing distributions and then an explanation of variation was given by dividing the distribution variance into three components (randomness, liability and proneness).