Analysis — RlFFENBURGH and CLUNIES-ROSS 
253 
Consider apPj + /3qPn- 
(3.1) 
+ /J qP n ) 
iTS 
■ ■+. HR31 
• o- V(2 tt) 
Equating the derivative to zero and rearrang- 
ing, we obtain 
which is a quadratic in c with minisum c as 
roots : 
(3.3;): c(ms) 
r^[v n 1 - <r i m 2 
r i' T 2 ^ m 2 - m 1 ) 2 - 2 ( or 2-“'i 
Equation (3.3) has three possibilities: 
( 1 ) when there are no real roots, 
(2) when no roots fall in (mi,m 2 ), 
and 
(3) when one and only one root falls 
in (rrii,m 2 ) . 
If a root should fall at one of mi,m 2 , this may 
be considered as a limiting case of situation 
(2). Situation ( 1 ) is trivial; all individuals are 
classified into one population. In situation ( 2 ) , 
linear discrimination is not very helpful; quad- 
ratic discrimination is indicated. In these situa- 
tions, possibly ( depending on parameters ) there 
is no discrimination which will be much of an 
improvement over the classification of all in- 
dividuals into one population or a purely ran- 
dom classification. Thus, situation (3) will be 
considered in this paper. 
When a root falls in (mi,m 2 ) , this is the 
root which minimizes apPj + /3qPn, and is 
therefore the root desired. The other root max- 
imizes apPi + /?qPn and therefore will not be 
used. Since a 2 has been arranged to be greater 
than o-i, and the smaller root is less than mi, 
the positive square root is required. When 
= cr 2 , c (ms ) is the root in (mi,m 2 ); the other 
root is infinite. 
Consider now the minimizing max (apPj, 
/3qPlI ) . apPj and /3qPn are monotonic, decreas- 
ing and increasing respectively, in c; and, there- 
fore, the desired c is located such that apPj = 
/3qPjI. An explicit result will not be found in 
general, since the integrals have not been eval- 
uated explicitly. If ap =r /3q, we have the in- 
tegrals identical except for upper limits of in- 
tegration, and apPj = /3qPji reduces to 
Solving, we obtain a minimax c: 
(3.5) c(mx) = - 1 2 LI 
. °2 + <r i 
It should be noted that if — o- 2 and ap 
=r /?q, both c(ms) and c(mx) reduce to a c 
dependent upon only the centroids, 
This c(m) is the population analogue of the 
c introduced for samples by Barnard (1935) 
and Fisher (1936) and currently used in linear 
discriminant analysis. 
4 . A Discussion of Levels and As 
The results (3.3) and (3.5) apply for the 
case in which loss functions and prior prob- 
abilities are known, i.e., (1.1). When either or 
both of these quantities are unknown, cor- 
responding to (1.2), (1.3), or (1.4), the cor- 
responding error quantities considered are given 
by (1.6), (1.7), or (1.8) respectively. The re- 
sults corresponding to (3.3) and (3.5) are ob- 
tained readily by the following substitutions in 
(3.3) and (3.5): 
(1.2) 'prior probabilities only”: a = ft = 1 
(1.3) "loss functions only”: p = q = 1 
(1.4) "neither”: a = i fi = p = q=l. 
For level (a), where both prior probabilities 
and loss functions are known, the risk may be 
measured and specified. If the total risk is to 
be minimized, then c(ms) is the appropriate 
constant. If the risk is to be minimized, subject 
to the restriction that risks from each source 
are to be equal, then c(mx) is the appropriate 
constant. 
For level (b), where prior probabilities only 
are known, then c ( ms ) minimizes the condi- 
tional probability of misclassification. However, 
if classification is only part of the problem at 
