Award Date


Degree Type


Degree Name

Doctor of Philosophy in Mathematical Sciences


Mathematical Sciences

Advisor 1

Malwane M. A. Ananda, Committee Chair

First Committee Member

Chih-Hsiang Ho

Second Committee Member

Hokwon Cho

Third Committee Member

Sandra Catlin

Graduate Faculty Representative

Chad Cross

Number of Pages



In this dissertation, we discuss the usability and applicability of three statistical inferential frameworks--namely, the Classical Method, which is sometimes referred to as the Conventional or the Frequentist Method, based on the approximate large sample approach, the Generalized Variable Method based on the exact generalized p -value approach, and the Bayesian Method based on prior densities--for solving existing problems in the area of parametric estimation. These inference procedures are discussed through Pareto and exponential distributions that are widely used to model positive random variables relevant to social, scientific, actuarial, insurance, finance, investments, banking, and many other types of observable phenomena. Furthermore, several Pareto and exponential populations, and the combination of several Pareto and exponential distributions are widely used in the Computer Networking and Data Transmission to model Self-Similar (SS) or Lthe ong-Range-Dependent (LRD) network traffic that can be generated by multiplexing several Pareto and exponentially distributed ON/OFF sources. One of the problems of interest in this dissertation is statistical inferences concerning common scale and common shape parameters of several Pareto distributions, and common location and common shape parameters of several exponential distributions based on the generalized p -value approach introduced by Tsui and Weerahandi where traditional frequentist or classical approaches do not provide useful solutions for the problems in the face of nuisance parameters. In this regard, we have developed exact tests and confidence intervals for common scale and common shape parameters of Pareto populations, and common location and common shape parameters of several exponential populations using ideas of generalized p -values and generalized confidence intervals. The resulting procedures are easy to compute and are applicable to small samples. We have also compared this test to a large sample test. Examples are given in order to illustrate results. In particular, using examples, it is pointed out that simply comparing classical and generalized p -values can produce a different conclusion that generalized pivotal quantities and generalized confidence intervals have proved to be very useful tools for making inference in practical problems. Furthermore, the Bayesian approach for the above problem is presented using the Gibbs sampling technique when shape parameters of several Pareto distributions and scale parameters of several exponential distributions are unknown. Their outcomes are compared with results based on classical and generalized approaches. The generalized inferential results derived for several Pareto and exponential populations are utilized extensively in finding exact solutions, as opposed to approximate solutions, for complicated functions of parameters of Pareto and exponential populations that are found in Computer Networking and Data Transmission. The Offered Optical Network Unit Load (OOL), which is a direct result of the transmission of data files, generated at the Optical Network Units (ONUs) is discussed at length, through various aspects of inferential techniques, to find exact and non-misleading solutions to provide attractive, fast, reliable, and sophisticated online service to the customers. Network traffic flows generated by Hyper Text Transfer Protocol (HTTP), File Transfer Protocol (FTP), Variable-Bit-Rate (VBR), and Video Applications are injected into the system to simulate the system. Most of the simulations and real experiments described in this dissertation were performed with the self-similar traffic. The self-similar traffic is generated by aggregating the cumulative packet count at a certain time of multiple substreams, each consisting of alternating Pareto ON - and OFF -periods or exponentially distributed ON -and OFF -periods. These periods modeled by the fractional Brownian motion or the fractional Gaussian noise exhibit a time series whose process is characterized by the stochastic process. Detailed statistical inferences based on the classical framework, the generalized framework, and the Bayesian framework for the Offered Optical Network Unit Load (OOL) and the other related Computer Networking physical quantities are discussed. Examples are given through real data in order to illustrate the newly introduced the Generalized Variable Method procedure. A limited simulation study is given to demonstrate the performance of the proposed procedure.


Bayesian method; Classical method; Data traffic; Exponential populations; Generalized variable method; Pareto distributions; Statistical inference; Statistical methods


Applied Mathematics | Applied Statistics | Statistical Methodology