Date of Award

8-1-2011

Degree Name

Doctor of Philosophy

Department

Educational Psychology

First Advisor

Headrick, Todd

Abstract

The Burr families (Type III and Type XII) of distributions are traditionally used in the context of statistical modeling and for simulating non-normal distributions with moment-based parameters (e.g., Skew and Kurtosis). In educational and psychological studies, the Burr families of distributions can be used to simulate extremely asymmetrical and heavy-tailed non-normal distributions. Conventional moment-based estimators (i.e., the mean, variance, skew, and kurtosis) are traditionally used to characterize the distribution of a random variable or in the context of fitting data. However, conventional moment-based estimators can (a) be substantially biased, (b) have high variance, or (c) be influenced by outliers. In view of these concerns, a characterization of the Burr Type III and Type XII distributions through the method of L-moments is introduced. Specifically, systems of equations are derived for determining the shape parameters associated with user specified L-moment ratios (e.g., L-Skew and L-Kurtosis). A procedure is also developed for the purpose of generating non-normal Burr Type III and Type XII distributions with arbitrary L-correlation matrices. Numerical examples are provided to demonstrate that L-moment based Burr distributions are superior to their conventional moment based counterparts in the context of estimation, distribution fitting, and robustness to outliers. Monte Carlo simulation results are provided to demonstrate that L-moment-based estimators are nearly unbiased, have relatively small variance, and are robust in the presence of outliers for any sample size. Simulation results are also provided to show that the methodology used for generating correlated non-normal Burr Type III and Type XII distributions is valid and efficient. Specifically, Monte Carlo simulation results are provided to show that the empirical values of L-correlations among simulated Burr Type III (and Type XII) distributions are in close agreement with the specified L-correlation matrices.

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