The q-cluster distribution

The paper studies the three-parameter generalization of the logarithmic distribution that is obtained as the cluster distribution for the generalized Euler distribution. The diagnostic statistic, R ( x ) = xp x / [ ( x - 1 ) p x - 1 ] , is constant for the logarithmic distribution. For the new distribution it can decrease, stay constant, or increase as x increases. The relative values of the extra parameters determine the flatness/hollowness of the distribution and its tail behaviour. Kempʹs q-logarithmic distribution and the Euler cluster distribution are special cases. Fitted data sets illustrate the properties of the distribution and its limiting forms.