Selection of the Linear and the Quadratic Discriminant Functions when the Difference between Two Covariance Matrices is Small

We consider selecting of the linear and the quadratic discriminant functions in two normal populations. We do not know which of two discriminant functions lowers the expected probability of misclassification. When difference of the covariance matrices is large, it is known that the expected probability of misclassification of the quadratic discriminant functions is smaller than that of linear discriminant function. Therefore, we should consider only the selection when the difference between covariance matrices is small. In this paper we suggest a selection method using asymptotic expansion for the linear and the quadratic discriminant functions when the difference between the covariance matrices is small.