For the classification problem between two normal populations with a common covariance matrix, we consider a class of discriminant rules based on a general discriminant function $T$. The class includes the one based on Fisher’s linear discriminant function and the likelihood ratio rule. Our main purpose is to derive an optimal discriminant rule by using an asymptotic expansion of misclassification probability when both the dimension and the sample sizes are large. We also derive an asymptotically unbiased estimator of the misclassification probability of $T$ in our class.
"An optimal discriminant rule in the class of linear and quadratic discriminant functions for large dimension and samples." Hiroshima Math. J. 34 (2) 231 - 250, July 2004. https://doi.org/10.32917/hmj/1150998164