Term of Award

Summer 2012

Degree Name

Master of Science in Mathematics (M.S.)

Document Type and Release Option

Thesis (open access)


Department of Mathematical Sciences

Committee Chair

Broderick O. Oluyede

Committee Member 1

Charles Champ

Committee Member 2

Hani Samawi


Author's abstract: In this thesis, a new class of weighted generalized beta distribution of the second kind (WGB2) is presented. The construction makes use of the conservability approach' which includes the size or length-biased distribution as a special case. The class of WGB2 is used as descriptive models for the distribution of income. The results that are presented generalize the generalized beta distribution of second kind (GB2). The properties of these distributions including behavior of pdf, cdf, hazard functions, moments, mean, variance, coefficient of variation (CV), coefficient of skewness (CS), coefficient of kurtosis (CK) are obtained. The moments of other weighted distributions that are related toWGB2 are obtained. Other important properties including entropy (generalized and beta), which are measures of the uncertainty in this class of distributions are derived and studied. Top-sensitive index, bottom-sensitive index, mean logarithmic deviation (MLD) index and Theil index obtained from generalized entropy (GE) are also applied practically. Dagum distribution is a special case of GB2, properties of Dagum and Weighted Dagum distributions including hazard function, reverse hazard function, moments are presented. Fisher information matrix (FIM) and estimates of model parameters under censoring including progressive Type II for the Dagum distribution are presented. WGB2 proved to be in the generalized beta-F family of distributions, and maximum likelihood estimation (MLE) is used to obtain the parameter estimates. WGB2 is applied as descriptive models for the size distribution of income, and fitted to U.S. family income (2001- 2009) data with different values of parameters. The empirical results show the length-biased distribution provides the best relative fit.

Included in

Mathematics Commons