252 
PACIFIC SCIENCE, Vol. XIV, July I960 
The first of these was introduced on level ( a ) by 
Brown (1950) and the second introduced on 
level (b) by Welch (1939). There has been 
more recent work on discriminant analysis, some 
of which is at levels similar to this treatment, 
but little seems applicable as the risk functions 
are not well defined. 
Each of these specifications leads to the choice 
of one out of a family of quadratic discrimina- 
tors. However, there are two related major dif- 
ficulties: one is the determination of which 
member of the family is appropriate (for the 
minimax solution), and the other is that the 
integrals giving Pj and Pn cannot be evaluated 
explicitly (for either solution), and no tables 
are available for the resulting Pi and Pip 
If the variance-covariance matrices of the two 
populations are equal, the quadratic discrimi- 
nator reduces to a linear discriminator; the in- 
tegrals for Pi and Pn may then be reduced to 
the incomplete integral of the standard normal 
density. This is always true for any linear dis- 
criminator. 
If we let A be a row vector of direction num- 
bers, X be a row vector of variables (represent- 
ing the possible measurements on the indi- 
vidual), c be a constant, and let primes denote 
transposition, then a linear discriminator may 
be written: 
(1.9) AX' = c. 
We lose no generality if we number the popu- 
lations such that the individual represented by X 
is classified into population I if AX' < c and 
into population II if AX' c. 
Let (mi, cri 2 ), (m 2 , cr 2 2 ) be the mean and 
variance of AX' when X is distributed as in 
populations I, II, respectively. Then it follows, 
using an obvious notation, that: 
A is well known (see, for example, Fisher, 
1936), being the inverse of this common matrix 
multiplied by the vector of difference means. 
When the variance-covariance matrices are not 
equal but are proportionate, then the correspond- 
ing A ( using either of the matrices ) is still op- 
timum under both the minisum and minimax 
criteria. 
In many fields the assumption of propor- 
tionate but not necessarily equal variance- 
covariance matrices is not unreasonable. This 
situation occurs, for example, in marine biology. 
The Hawaiian tunas ahi ( Neothunnus macrop- 
terus ) and ahipahala ( Thunnus alalunga ) are 
similar in most respects, but the ahi is a larger 
and more complex fish. If weight, fork length, 
lengths of second dorsal and anal fins, and the 
ratio of the length of the pectoral fin to the 
fork length (which varies inversely as the first 
four variables ) are taken to be the variables, the 
population variance-covariance matrices for the 
ahi and ahipahala are (expected to be) propor- 
tional but unequal. Another example is cited in 
the literature, although only two variables were 
used. Mottley ( 1941 ) found that the variances 
and covariance for head and body measurements 
of trout ( Salmo gairdnerii kamloops) stocked 
in two Canadian lakes were proportional. 
The optimum A for general dispersion ma- 
trices is not easy to derive. This problem is con- 
sidered in another paper by the authors ( I960) . 
The current paper considers optimum c for 
given A and thus in what follows it is only 
necessary to consider that A has been deter- 
mined either by the methods mentioned above 
or arbitrarily. 
3. The Constant c for Minimized Error 
Quantities 
(1.10) P I 
N(0 , 1) dx: 
( 1 . 11 ) 
2. The Appropriate Linear Function 
For the case when the distributions have 
identical variance-covariance matrices, the vector 
We lose no generality if we let m 2 > mi 
and a 2 ^ o-i. The designation of the population 
having the larger standard deviation as popula- 
tion II is arbitrary. We may then make a scale 
transformation of ± 1, whichever is necessary 
to obtain m 2 > mi. 
We now wish to obtain expressions for the 
constant c which will minimize the error quan- 
tities under the minisum and minimax criteria, 
respectively. 
