A bootstrap generalized likelihood ratio test in discriminant analysis

A generalized likelihood ratio test is developed for classification in two populations when one needs to control one of the probabilities of misclassification. The proposed classification procedure is constructed by applying the parametric bootstrap to the generalized likelihood ratio. There are known methods for controlling this misclassification probability for the case where normal distributions with the same covariance matrix are assumed. Our approach, however, can be applied to not only this case but to the case of normal distributions with different covariance matrices and the case of a mixture of discrete and continuous variables. The results given here do not depend on normality but can, in fact, be applied to any distribution for which the maximum likelihood estimates exist. We do, however, restrict our simulation of these results to the normal distribution if the variates are all continuous. Three cases are simulated: normal distributions with equal covariance matrix, normal distributions with unequal covariance matrices, and mixture of categorical and normal variables, An application to classifying seismic events is presented.