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.