Optimal sample allocation for normal discrimination and logistic regression under stratified sampling

For two multivariate normal populations with a common covariance matrix and stratified sampling, we consider two methods of estimation—Fisher’s linear discriminant function and logistic regression. Intuition suggests that taking half of the observations from each population is a reasonable design choice. Based on minimizing the expected error regret, asymptotic optimal sample allocations are found. The results indicate that the differences in the expected error regret for optimal versus balanced allocation are generally quite small. It is recommended that equal sample sizes for the two populations be used for these problems.