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Comparison of the probabilities of misclassification for the estimated linear, quadratic, and unbiased density discriminant functions using asymptotic expansions
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Comparison of the probabilities of misclassification for the estimated linear, quadratic, and unbiased density discriminant functions using asymptotic expansions
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Comparison of the probabilities of misclassification for the estimated linear, quadratic, and unbiased density discriminant functions using asymptotic expansions
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Dissertation
Comparison of the probabilities of misclassification for the estimated linear, quadratic, and unbiased density discriminant functions using asymptotic expansions
1988
Overview
Standard discriminant functions tend to perform well only with large training samples. This study developed a theoretical foundation for deriving the asymptotic expansions of the probabilities of misclassification for a broad class of discriminant functions. Three of these were evaluated: the linear discriminant function (LDF), the quadratic discriminant function (QDF), and the unbiased-density discriminant function (UDF). The UDF is based on the UMVU estimator of the multivariate normal density function and is asymptotically equivalent to the QDF. A class of polynomials with multiple matrix arguments as well as concepts of discrimination potential, efficiency, and deficiency were defined. Under the assumption of equal covariances in multivariate normal populations the deficiencies of the QDF and UDF relative to the LDF indicated that both functions are inferior to the LDF and that the UDF is always inferior to the QDF. An evaluation of efficiencies with smaller sample sizes showed that when the distance between populations is small, the relative efficiencies of the QDF and UDF are quite poor compared to the LDF with the UDF always slightly worse. Under the assumption of unequal covariances, the UDF continued to perform poorly while the QDF outperformed the LDF only for large sample sizes and large differences in covariances. An application to medical diagnosis using clinical laboratory test was studied. The main conclusions from the research were that the UDF is inferior to the QDF and that optimal density estimation does not improve the discriminant function under the conditions studied. An approach that deserves further research is the optimal estimation of the partition between population. The field of discriminant analysis needs a firmer theoretical basis instead of the current empirical approaches.
