FOUNDATIONS OF THEORETICAL STATISTICS. 
331 
The region d0dO x evidently lies wholly in the isostatistical region do. 
equation 
4 lo g/( 0 > °i) = 0 
uu 
Hence the 
is satisfied, irrespective of 6 1 , by the value 0 = 0. This condition is satisfied if 
f(0, 0, 0 X ) = (j> (0, S) . 0 X ); 
for then 
l log/= l log *- 
and the equation for the optimum degenerates into 
log 0 ) = 0, 
which does not involve 0 X . 
But the factorisation of / into factors involving (0, $) and (0, 0 X ) respectively is merely 
a mathematical expression of the condition of sufficiency; and it appears that any 
statistic which fulfils the condition of sufficiency must be a solution obtained by the 
method of the optimum. 
It may be expected, therefore, that we shall be led to a sufficient solution of problems 
of estimation in general by the following procedure. Write down the formula for the 
probability of an observation falling in the range dx in the form 
f (0, x) dx, 
where 0 is an unknown parameter. Then if 
L = S (log/), 
the summation being extended over the observed sample, L differs by a constant only 
from the logarithm of the likelihood of any value of 0. The most likely value, 0, is 
found by the equation 
and the standard deviation of 0, by a second differentiation, from the formula 
&L _ J_. 
0<9 2 <r« 2 ’ 
this latter formula being applicable only where 0 is normally distributed, as is often 
the case with considerable accuracy in large samples. The value a-g so found is in 
these cases the least possible value for the standard deviation of a statistic designed to 
