Standard Errors Of Discriminant Function Coefficients In The Presence Of An Additional Partial Information (missing Observations, Maximum Likelihood Estimation, Fisher's).

This dissertation deals with the derivation of the standard errors of the coefficients in a discriminant function between two populations, when in addition to complete samples from the two populations, one has a sample from one of the populations on a subset of the variables only. The maximum likelihood estimates (M.L.E.) of the means of the two populations and the common variance-covariance matrix are first derived. Using these, the M.L.E.'s of the discriminant function coefficients are derived. These require the estimates of the inverse of the variance-covariance matrix. For deriving the standard errors, several methods are presented. Exact results are given in the bivariate case. These are then generalized to the p-variate case. This generalization is achieved through the use of vectorization of a matrix and the related Vec operator. Properties of Wishart matrices and inverses of Wishart matrices, as derived by Haff (1979), Magnus and Neudecker (1979), are used repeatedly for this purpose. Almost exact results have been obtained in this case also. The delta technique of deriving asymptotic variances and covariances is also presented. In addition, the information theory approach of Smith and Hocking (1968) for the estimated variance-covariance matrix is extended to its inverse in this dissertation. Information matrices from independent observations are additive but variance-covariance matrices are not. Fisher's information matrix is the inverse of the large sample variance-covariance matrix of the M.L.E.'s of the parameters. Smith and Hocking use this concept to get the large sample variances and covariances, and we have extended this to the discriminant function coefficients. The decrease in the variances of the discriminant function coefficients due to the use of the extra partial information is examined in this dissertation. This decrease is a measure of the gain in efficiency resulting from the additional observations. Some numerical values are computed and presented in order to assess this gain in precision. Finally, the problem of estimation of the mean vectors and the variance-covariance matrix of more than two populations with extra partial information is considered briefly.Ph.D.Biological SciencesBiostatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/127806/2/8600459.pd