Parameter Estimation of the Mixed Generalized Gamma Distribution Using Maximum Likelihood Estimation and Minimum Distance Estimation

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.

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Document Details

Document Type
Technical Report
Publication Date
Mar 01, 1998
Accession Number
ADA342390

Entities

People

  • Dean G. Boerrigter

Organizations

  • Air Force Institute of Technology

Tags

Communities of Interest

  • C4I
  • Energy and Power Technologies
  • Space
  • Weapons Technologies

DTIC Thesaurus Topics

  • Air Force
  • Algorithms
  • Basic Programming Language
  • Computer Programs
  • Distribution Functions
  • Engineering
  • Equations
  • Estimators
  • Genetic Algorithms
  • Information Science
  • Maximum Likelihood Estimation
  • Monte Carlo Method
  • Normal Distribution
  • Probability Density Functions
  • Probability Distributions
  • Random Variables
  • Statistical Analysis

Fields of Study

  • Mathematics

Readers

  • Statistical inference.

Technology Areas

  • AI & ML
  • AI & ML - Bayesian Inference
  • Biotechnology