Date of Award
Doctor of Philosophy (PhD)
Mathematics and Statistics
Computational and Applied Mathematics
N. Rao Chaganty
Dayanand N. Naik
Michael J. Doviak
Analysis of dependent continuous and discrete data has become an active area of research. For normal data, correlations fully quantify the dependence. And historically, maximum likelihood method has been very successful to estimate the correlations and unbiased estimating equation approach has become a popular alternative when there may be a departure from normality. In this thesis we show that the optimal unbiased estimating equation coincides with the likelihood equations for normal data. We then introduce a general class of weighted unbiased estimating equations to estimate parameters in a structured correlation matrix. We derive expressions for asymptotic covariance of the estimates, and use those expressions to determine the optimal weights. We also study an important subclass of unbiased estimating equations. The optimal weights for this subclass are not tractable, especially for the familial correlation structure. We suggest approximations and study performance of these approximate weights using simulations.
For familial binary responses we first investigate ranges of associations measures, which include odds ratios, kappa statistics, and relative risks besides correlations. Knowing and understanding these ranges is important for developing efficient estimation methods. We study estimation of the familial correlations using a probit model and stochastic representation of the latent variables. We discuss some extensions of our results to nuclear families. Some real life examples are presented to illustrate the estimation methods.
"Efficient Unbiased Estimating Equations for Analyzing Structured Correlation Matrices"
(2006). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/htxp-1s85