346 
MR. R. A. FISHER ON THE MATHEMATICAL 
sary as soon as we take a view unrestricted by the method of moments ; of the so-called 
heterotypic curves between r — 3 and r = 7 it should be noticed that they not 
only fall into the ordinary Pearsonian types, but have finite values for the moment 
coefficients and ft., ; they differ from those in which r exceeds 7, merely in the fact 
that the value of fi 2 , calculated from the fourth moment of a sample, has an infinite probable 
error. It is therefore evident that this is not the right method to treat the sample, but 
this does not constitute, as it has been called, “ the failure of Type IV.,” but merely 
the failure of the method of moments to make a valid estimate of the form of these 
curves. As we shall see in more detail, the method of moments, when its efficiency is 
tested, fails equally in other parts of the diagram. 
In expression (3) we have found that the efficiency of the method of moments for 
location of a curve of Type IV. is 
E = i'- l (/- + 4' + i--) ^ 
r- FI r + 2 r + 4 (r 2 4- v 2 ) 
whence if we substitute for r and v in terms of the co-ordinates of our diagram, we obtain 
a general formula for the efficiency of the method of moments in locating Pearsonian 
curves, which is applicable within the boundary of the zero contour (fig. 3). This may 
Fig. 3. Region of validity of the first moment (the mean) applied in the location of 
Pearsonian curves showing contours of efficiency. 
be called the region of validity of the first moment ; it is bounded at the base by the 
line r = 1, so that the first moment is valid far beyond the heterotypic limit; its other 
boundary, however, represents those curves which make a finite angle with the axis at 
the end of their range (m x , or m 2 , = 1) ; all J curves (m v or m.,, < 0) are thus excluded. 
This boundary has a double point at P. which thus forms the apex of the region of validity. 
