Term of Award

Summer 2011

Degree Name

Master of Science in Mathematics (M.S.)

Document Type and Release Option

Thesis (restricted to Georgia Southern)

Department

Department of Mathematical Sciences

Committee Chair

Broderick O. Oluyede

Committee Member 1

Charles Champ

Committee Member 2

Hani Samawi

Committee Member 3

n/a

Abstract

Weighted distributions occur naturally and frequently in research related to reliability, bio-medicine, ecology and in the modeling of clustered sample, heterogeneity, and extraneous variation. In the analysis of intervention data, the expected value of the duration to completion of a random event sampled randomly at the end of its duration turns out to be approximately equal to the expected duration of its random interventions. This is due to the concept of size or length-biased sampling, where the weight function represents the duration of the random event in the life cycle assessment. The case of chronic disease identified by early detection screening programs constitute a length biased (weighted) sampling procedure, as individuals with a long pre-clinical disease phase have a greater probability of being identified. In this research, a new weighted generalization of the Raleigh distribution is constructed. The construction makes use of the "conservability approach" which includes the size or length-biased distribution as a special case. Weighted generalized Raleigh distribution (WGRD) with several weight functions are constructed. The properties of these distributions including behavior of hazard or failure rate and reverse hazard functions, moments, moment generating function, mean, variance, coefficient of variation, coefficient of skewness, coefficient of kurtosis are obtained. Other important properties including entropy (Shannon, beta and generalized) which are measures of the uncertainty in these distributions, and Fisher information which measures the amount of information that a random variable carries about the distribution's unknown parameters are derived and studied. Estimation of the parameters of the weighted generalized Raleigh distribution including the maximum likelihood estimators are derived. Test procedures for weightedness including length-biasedness concerning the Raleigh, generalized Raleigh and weighted generalized Raleigh models are developed.

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