On the reparameterization of models to obtain uncorrelated maximum likelihood estimators
Walter, Loraine D.
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This thesis is concerned primarily with the development of criteria and methods for the reparameterization of models and distributions in order that independent parameter estimators be obtained. The criteria for independence are required not to be functions of the data, which enables their use before estimation of the parameters. Methods for the application of these criteria are developed for this type of use and involve the specification of a family of parameterizations of the given model from which that member(s), if any, is to be selected which satisfies one of the criteria. The establishment of general measures of normality which can be used to indicate how well the estimator distributions for selected parameterizations are approximated by the normal is also desired. The approach taken in this work is to suggest criteria, test them on simple cases, and then apply them to more sophisticated models (of at most two parameters) for the purposes of developing techniques and discovering problem areas. Three criteria for independence which do not require knowledge of the estimator distribution or its moments are suggested: zero asymptotic correlation, zero mixed second derivative of the log likelihood, and each first derivative of the log likelihood a function of one parameter only; the one used in any situation is to be determined by properties of the particular model. These are applied to three models: the two-parameter bivariate exponential distribution, the Gamma distribution, and the bioassay logistic model. The first of these is uninteresting except as a test of the criteria. For the Gamma distribution it was found that parameters which are simple functions of the original parameters satisfy the criterion of zero asymptotic correlation. Several sets of Gamma data are generated for a few parameter values and the estimates of asymptotic correlation obtained from these are compared for estimators of the two parameterizations; those for the new parameters are close to zero and uniformly less than those for the original parameters. (Chapter 3) Parameterization of the bioassay logistic model is studied in considerable depth. (Chapters 4 and 5)