FOUNDATIONS OF THEORETICAL STATISTICS. 
Q 
O 
5 J 
while that of the mean of the extreme values is 
2 5 
a 
so yielding a ratio 3 /n. It is probable that this quantity may prove a suitable substitute 
for the efficiency of a statistic for curves beyond its region of validity. 
To determine the efficiency of the moment coefficients /3i and /3 2 in determining the 
form of a Pearsonian curve, we must in general apply the method of Section 8 to the 
calculation of the simultaneous distribution of the four parameters of those curves when 
estimated by the method of maximum likelihood. Expressing the curve by the formula 
appropriate to Type IV., we are led to the determinant 
r + 1r + 2r + 4 
a 2 (r + 4 +v 2 ) 
r +1 r + 2 v 
a 2 (r + 4 + p 2 ) 
r +1 r + 2 v 
a 2 (r+4 + v 2 ) 
r+l (2-r + 4 + p 2 ) 
a 2 (7+4 +p 2 ) 
r+lr+2 
a (r + 2~ + v 2 ) 
r + 1 v 
ct (r + 2 + v") 
r +1 v 
a (r + 2 + v 2 ) 
r+ 2 + v“ 
a (r + 2 2 + j/ 2 ) 
r+ 1 r+2 
a (r + 7 + v) 
r+ 1 v 
a(r + 2 +v 2 ) 
r +1 v 
ci ( r + 2 + p 2 ) 
r + 2 + v 2 
a (r + 2" + v 2 ) 
S 2 
- log F 
OP 
log F 
^2 
C 
3 
ov 
or 
log F 
as the Hessian of —L, when 
F = e 
e” e sin r 6 cl6. 
The ratios of the minors of this determinant to the value of the determinant give 
the standard deviations and correlations of the optimum values of the four parameters 
obtained from a number of large samples. 
In discussing the efficiency of the method of moments in respect of the form of the 
curve, it is doubtful if it be possible to isolate in a unique and natural manner, as we 
have done in respect of location and scaling, a series of parameters which shall successively 
represent different aspects of the process of curve fitting. Thus we might find the 
efficiencies with which r and v are determined by the method of moments, or those of 
the parametric functions corresponding to and /T, or we might use m ] and m 2 as 
independent parameters of form ; but in all these cases we should be employing an 
arbitrary pair of measures to indicate tire relative magnitude of corresponding contour 
ellipses of the two frequency surfaces. 
For the symmetrical series of curves, the Types II. and VII., the two systems of 
3 c 2 
