FOUN DATIONS OF THEORETIC AT. 
STATISTICS. 
387 
from nn' observations will be determined with the same accuracy as the position, as a 
whole, of a curve of known form and size, can be determined from a sample of n observa¬ 
tions when n is large. Then 
d; 
but, when p is laroe, 
and 
therefore 
o I / ,\ 2 1 5 
2 y— log p!)- - + — 
dp p p 
d 2 
p + 1 g l0g {p 0 — 1 =-( 1-— + —2 • • 
1 dp 2 6 1 ■ 2 p \ 3 p 3 p 2 
2 Iqo- (p 1) _ — + — — J_ (i —I_ 
dp 2 * KP ' } p p 2 Sp 3 \ 5p 2 
n' = f p 2 ( I — ~~ p + 
2 
8 
3 p x I5p 2 
= f p 2 —p> + i- 
For large values of p the probable error of the determination of the end-point may be 
found approximately by multiplying the probable error of location by 
[p-¥> \/f- 
As p grows smaller, n' diminishes until it reaches unity, when p — 1. For values of 
p less than I it. would appear that the end-point had a smaller probable error than the 
probable error of location, but, as a matter of fact, for these values location is determined 
by the end-point, and as we see from the vanishing of <r A , whether or not p and a 
are known, whenp = 1, the weight of the determination from this point onwards increases 
more rapidly than n, as the sample increases. (See Section 10.) 
The above method illustrates how it is possible to calculate the variance of any 
function of the population parameters as estimated from large samples ; by comparing 
this variance with that of the same function estimated by the method of moments, we 
may find the efficiency of that method for any proposed function. The above examina¬ 
tion, in which the determinations of the locus, the scale, and the form of the curve are 
treated separately, will serve as a general criterion of the application of the method of 
moments to curves of Type III. Special combinations of the parameters will, however, 
be of interest in special cases. It may be noted here that by virtue of equation (2) the 
function of m -f- a (p -)- l) is the same,, whether determined by moments or by the method 
of the optimum : 
uiy + (p IJL + 1) = m + o (p + 1). 
The efficiency of the method of moments in determining this function is therefore 100 
per cent. That this function is the abscissa of the mean does not imply 100 per cent, 
efficiency of location, for the centre of location of these curves is not the mean (see p. 340). 
