STOCHASTIC ORDERINGS AND PARAMETER ESTIMATION FOR THE HURWITZ-LERCH ZETA DISTRIBUTION

Abstract

The Hurwitz-Lerch Zeta (HLZ) family includes many well-known distributions such as the logarithmic distribution, Zipf-Mandelbrot distribution and so on. In this study, the stochastic orderings of the random variables in the HLZ family are established based on the likelihood ratio. These orderings provide a useful tool for comparing the ‘magnitudes’ of these random variables. The tail probability of the HLZ distribution is shown to have an interesting relation with a generalisation of the logarithmic distribution (GLD) proposed in [1]. To demonstrate the flexibility of the HLZ distribution in empirical modelling, a robust probability generating function (pgf) based estimation method using Hellinger-type divergence is implemented in data-fitting and the results are compared with various other GLD’s. An augmented pgf is constructed to overcome the difficulties of this estimation method when some data are grouped. (Keywords: Generalised logarithmic distribution, Hellinger-type divergence, maximum likelihood, probability generating function based estimation, Zipf-Mandelbrot distribution)