Statistical modelling and inference for a class of bivariate and related distributions / Ng Choung Min - Ng, Choung Min (2010)
This thesis considers bivariate extension of the Meixner class of distributions by the method of generalized trivariate reduction so that the marginal distributions have different parameters; in particular, a new bivariate negative binomial (BNB) distribution is examined. Different marginal parameters allow flexibility in statistical modelling and simulation studies when different marginal distributions and a specified correlation are required. The multivariate extension of this class of distributions is also given. Specifically, various interesting properties of the proposed BNB distribution, such as canonical expansion and quadrant dependence are examined. In addition, potential applications of the proposed distribution, as a bivariate mixed Poisson distribution, and the computer generation of bivariate samples are discussed. Due to the complicated or intractable joint probability function (pf) for most bivariate and multivariate distributions, the popular method of maximum likelihood estimation (MLE) either leads to a slow parameter estimation or totally could not be employed. Furthermore, MLE is not robust in the presence of outliers. Alternative robust methods like minimum Hellinger distance (MHD) can be used but these methods may also involve complicated pf. To address this difficulty in estimation, a Hellinger type distance measure based on the probability or moment generating function is proposed as a tool for quick and robust parameter estimation. The proposed method is shown to yield consistent estimators. It is computationally much faster than MLE or MHD since the generating function required is usually much simpler compared to the corresponding pf. The distribution of the difference of two discrete random variables, particularly that of two correlated negative binomial random variables from the proposed BNB distribution, is also studied.The application of this distribution, which caters for non homogeneity in a group of individuals, in modelling fluctuating asymmetry based on meristic (counts) traits in organisms is discussed. A test for fluctuating asymmetry, based on a zero-inflated count model, is examined. Also, numerical illustrations are given to complement the ideas and theories put forth.