332 
MR. R. A. FISHER ON THE MATHEMATICAL 
estimate the same parameter ; it may therefore be applied to calculate the efficiency of 
any other such statistic. 
When several parameters are determined simultaneously, we must equate the second 
differentials of L, with respect to the parameters, to the coefficients of the quadratic 
terms in the index of the normal expression which represents the distribution of the 
corresponding statistics. Thus with two parameters. 
8 2 L = 1 J_ 
a ^ 2 
<pl = i j_ 
ad 2 2 i-4'4’ 
a 2 L _ I r 
30 2 1 — r 2 ^., a 
or, in effect, erf is found by dividing the Hessian determinant of L, with respect to the 
parameters, into the corresponding minor. 
The application of these methods to such a series of parameters as occur in the speci¬ 
fication of frequency curves may best be made clear by an example. 
8. The Efficiency of the Method of Moments in Fitting Curves of the 
Pearsonian Type III. 
Curves of Pearson’s Type III. offer a good example for the calculation of the efficiency 
of the Method of Moments. The chance of an observation falling in the range dx is 
df = 
a 
pi 
x—m 
a 
V _ x ~~ m 
3 a dx* 
By the method of moments the curve is located by means of the statistic its dimen¬ 
sions are ascertained from the second moment /x 2 , and the remaining parameter p is 
determined from /3i- Considering first the problem of location, if a and p were known 
and we had only to determine m, we should take, according to the method of moments, 
Mi — nifj. +ct (p + 1), 
where m M represents the estimate of the parameter m, obtained by using the method of 
moments. The variance of is, therefore, 
2 2 m 3 a 2 {p+l) 
= <T n, = — = - - - • 
If, on the other hand, we aim at greater accuracy, and make the likelihood of the 
sample a maximum for variations of m, we have 
L = 
—n log a—n log {pl)+pS^ log -—— ) 
* The expression, x !, is used here and throughout as equivalent to the Gaussian II ( x ), or to T (a:+l), 
whether x is an integer or not. 
