FOUNDATIONS OF THEORETICAL STATISTICS. 
•>.n 
57 
distributions are reduced to discontinuous distributions, and in an exact discussion must 
be treated as such. 
If p s be the probability of an observation falling in the cell (s), r p s being a function of 
the required parameters 0 ,, eu... ; and in a sample of N, if n s are found to fall into 
that cell, then 
S (log/) = S (n, lo gp s ). 
If now we write n s = p s N, we may conveniently put 
L = S ( n, log — j > 
‘ nj 
where L differs by a constant only from the logarithm of the likelihood, with sign 
reversed, and therefore the method of the optimum will consist in finding the minimum 
value of L. The equations so found are of the form 
3L 
00 
-sf 
\n 
n Q otia 
00 / 
0. 
( 6 ) 
It is of interest to compare these formulae with those obtained by making the Pearsonian 
X 2 a minimum. 
For 
2 _ a {n s -n s Y 
X — & -’ 
n s 
and therefore 
1+x 2 
5 
so that on differentiating bv do, the condition that / should be a minimum for variations 
of 0 is 
= 0 
(7) 
Equation (7) has actually been used ( 12 ) to “ improve ” the values obtained by the 
method of moments, even in cases of normal distribution, and the Poisson series, where 
the method of moments gives a strictly sufficient solution. The discrepancy between 
these two methods arises from the fact that / is itself an approximation, applicable 
only when n s and n s are large, and the difference between them of a lower order of 
magnitude. In such cases 
L = S 
and since 
VOL. ccxXII.—A. 
i n c\ a l - 1 m + x 
n s log zr ) = o \m-\-x fog- 
m 
>S \x+ - - 7 U,. 
2m 6 m 
S (x) = 0 , 
3 D 
