ABSTRACT
The quantity a is called the Mahalanobis distance between two normal populations with the same covariance matrix. Now the minimax classification rule R is given by, writing u=U(x),
where the constant k is given by
or, equivalently, by
where Suppose we have a group of N individuals, with observations xa, a=1,…, N, to be
classified as a whole to one of the pi, i=1, 2. Since, writing
and is normally distributed with means Na/2, - Na/2 and the same variance Na under p1 and p2, respectively, the Bayes classification rule R=(R1, R2) against the a priori probabilities (p1, p2) is given by
The minimax classification rule R=(R1, R2) is given by (9.25), where k is determined by
If the parameters are unknown, estimates of these parameters are obtained from independent random samples of sizes N1 and N2 from p1 and p2, respectively. Let
a=1,…, N1, a=1,…, N2, be the sample observations (independent) from p1, p2, respectively, and let
We substitute these estimates for the unknown parameters in the expression for U to obtain the sample discriminant function [?(x)]
which is used in the same way as U in the case of known parameters to define the classification rule R. When classifying a group of N individuals instead of a single one we can further improve the estimate of S by taking its estimate as s, defined by
The sample discriminant function in this case is
The classification rule based on ? is a plug-in rule. To find the cutoff point k it is necessary to find the distribution of V. The distribution of V has been studied by Wald (1944), Anderson (1951), Sitgreaves (1952), Bowker (1960), Kabe (1963),
and Sinha and Giri (1975). Okamoto (1963) gave an asymptotic expression for the distribution of V.
