Probabilistic properties and statistical inference for a family of generalised and related distributions / Liew Kian Wah - Liew, Kian Wah (2013)
Three generalised distributions are studied in this thesis from different aspects. The Hurwitz-Lerch zeta distribution (HLZD) that generalises the logarithmic distribution and a class of distributions that follows the power law is considered. To investigate the effects of parameters on the stochastic properties of the HLZD, stochastic orders between members in this large family are established. A relationship between the tail behaviours of the HLZD and that of a class of generalised logarithmic distribution is highlighted. The HLZD has shown good flexibilities in empirical modelling. A robust probability generating function based estimation method using Hellinger-type divergence is implemented in data-fitting and the results are compared with various other generalisations of logarithmic distribution. An augmented probability generating function is constructed to overcome the difficulties of this estimation procedure when some data are grouped. The Poisson-stopped sum of the Hurwitz-Lerch zeta distribution (Poisson-HLZD) is then proposed as a new generalisation of the negative binomial distribution. Several methods have been used in deriving the probability mass function for this new distribution to show the connections among different approaches from mathematics, statistics and actuarial science. Basic statistical measures and probabilistic properties of the Poisson-HLZD are examined and the usefulness of the model is demonstrated through examples of data-fitting on some real life datasets. Finally, the inverse trinomial distribution (ITD) is reviewed. Both Poisson-HLZD and ITD are proved to have mixed Poisson formulation, which extend the applications of the models for various phenomena. The associated mixing distribution for the ITD is obtained as an infinite Laguerre series and the result is compared to some numerical inversions of Laplace transform.