Linear dimension reduction and Bayes classification with unknown population parameters

Odell and Decell, Odell and Coberly gave necessary and sufficient conditions for the smallest dimension compression matrix B such that the Bayes classification regions are preserved. That is, they developed an explicit expression of a compression matrix B such that the Bayes classification assignment are the same for both the original space x and the compressed space Bx. Odell indicated that whenever the population parameters are unknown, then the dimension of Bx is the same as x with probability one. Furthermore, Odell posed the problem of finding a lower dimension q < p which in some sense best fits the range space generated by the matrix M. The purpose of this paper is to discuss this problem and provide a partial solution.