#### Date of Award

3-1998

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Department of Operational Sciences

#### First Advisor

Edward A. Pohl, PhD

#### Second Advisor

W. Paul Murdock, PhD

#### Abstract

The Generalized Gamma is an extremely flexible distribution that is useful for reliability modeling. Among its many special cases are the Weibull and Exponential distributions. A mixture of Generalized Gamma Distributions is even more useful because multiple causes of failure can he simultaneously modeled. This research studied parameter estimation of the special cases of the Mixed Generalized Gamma Distribution and built upon them until the full nine-parameter distribution was being estimated. First, special cases of a single Generalized Gamma Distribution were estimated. Next, mixtures of Exponential distributions with both known and unknown location parameters were estimated. Next, mixtures of Weibull distributions with both known and unknown location parameters were estimated. Lastly, the full nine- parameter Mixed Generalized Gamma Distribution was estimated. Two techniques were used to estimate the parameters of each distribution. The first technique used was the Method of Maximum Likelihood. The log likelihood equation was maximized using a Genetic Algorithm. The second technique used was the Method of Minimum Distance. This technique takes the Maximum Likelihood parameter estimate as initial estimate. With this initial estimate, the mixture and the first location parameter are sequentially varied to minimize the Anderson-Darling statistic between the estimated cumulative distribution function and the empirical distribution function. These two parameters are then fixed at their Minimum Distance values and the remaining parameters are re-estimated using Maximum Likelihood. Minimum Distance Estimation was demonstrated to improve the parameter estimates from Maximum Likelihood for almost all of the special case distributions tested.

#### AFIT Designator

AFIT-GOR-ENS-98M-03

#### DTIC Accession Number

ADA342390

#### Recommended Citation

Boerrigter, Dean G., "Parameter Estimation of the Mixed Generalized Gamma Distribution Using Maximum Likelihood Estimation and Minimum Distance Estimation" (1998). *Theses and Dissertations*. 5578.

https://scholar.afit.edu/etd/5578

## Comments

Co-advised thesis.