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PACIFIC SCIENCE, Vol. XIV, July I960 
hand, then it may be desirable, in order to avoid 
bias in later stages, say, to minimize, subject 
to equalizing the probabilities of the two types 
of misclassification; here c(mx) is the appro- 
priate constant. 
For example, consider a merchandizing situa- 
tion. If the problem is to allocate a limited ship- 
ment of goods to two branches of the same 
store, the same management suffers the loss 
from understocking either branch, and c(ms) 
is the appropriate constant to use in specifying 
the quantities of goods to go to each branch. 
On the other hand, if the problem is to equalize 
buyer-seller risk, as in the case of an inde- 
pendent mediator handling quality control, then 
c(mx) is the appropriate constant to use in 
specifying the acceptable level of quality. 
For levels (c) and (d), the error quantities 
are in no sense absolute quantities. Here c(mx) 
will be the most reasonable constant to use, 
since under the minimax solution the expected 
numbers of misclassifications are equal for the 
two populations. 
In practice, a, (3, p, and q may or may not be 
well defined conceptually, but either way will 
often, perhaps usually, be unknown. Thus a com- 
parison between discriminators using c(ms), 
c(mx), and c(m) at level (d) is appropriate. 
5. Comparison of Discriminators 
Introduction. The discriminators may be 
compared on the basis of our minisum and 
minimax criteria. Let us designate these criteria 
respectively in terms of the error quantities as 
(i) P s = Pi + Pll 
(ii) P x = max (Pi, Ph). 
In comparing discriminators, it can happen 
that either one has both criteria less than or 
equal to those of the other or this does not occur. 
If the former holds, then the discriminator with 
the smaller criteria may be said to be better 
than the other. This is true whether the dis- 
crimination is linear or not. 
For the purposes of this paper, A has been 
taken to be a vector of constants. Thus, while 
linear discriminators are functions of both A 
and c, our comparison need be concerned only 
with varying c’s. The restriction to level (d) 
together with the vector of constants, A, enables 
us to keep the number of parameters down to 
two for comparisons of the discriminators AX' 
= c ( ms ) , AX' = c ( mx ) , and AX' = c(m). 
c(ms) and c(mx) are the c’s derived for the 
two criteria; both reduce to c(m) in the special 
case of equal dispersion matrices, c ( m ) is the 
population analogue of the c used in practice 
and is easier to compute than are c(ms) and 
c ( mx ) . Since c ( mx ) andc(ms) each minimize 
one criterion, the comparisons will be to find 
the conditions under which c(m) leads to both 
smaller P x than does c ( ms ) and smaller P s than 
does c ( mx ) . When these conditions are satisfied 
then c(m) may be regarded as a compromise 
between c ( ms ) and c ( mx ) . 
The two essential parameters will be defined 
as 
(5.1) B- = °2/<r 
Tn - m 
r 1 
It can be seen that B ^ 1 and C > 0. If results 
in B and C should be tabulated, the tables would 
be symmetric in log B, — - log B, and in C, — • C. 
Condition for reasonable linear discrimination. 
Under certain conditions, linear discrimination 
does not yield good results; an example of this 
is the situation in which the centroids of the 
two populations are the same. Any description 
of the conditions necessary for linear discrimina- 
tion to be able to lead to reasonable results must 
be, to some extent, arbitrary. Generally, the sit- 
uations in which linear discrimination may be 
rejected are typified by no root of c(ms) being 
contained in (mi,m 2 ). 
At level (d) there are always two real solu- 
tions of (3.3). By restricting our interest to the 
range (mi, m 2 ) it follows from considerations 
of monotonicity, continuity, and limiting be- 
havior that a necessary and sufficient condition 
for the existence of a root of (3.3 ) in this range 
is 
since the left and right sides of the inequality 
are the densities of populations I and II at m 2 . 
(5.3) may be rewritten in terms of B and C 
as follows: 
(5.4) C > 2 (3 + l)' 2 In B 
