FOUNDATIONS OF THEORETICAL STATISTICS. 
355 
In all these functions and those of the following table, r must be substituted as a 
positive quantity, although it must not be forgotten that r changes sign as we pass from 
Type VII. to Type II., and we have hitherto adhered to the convention that r is to 
be taken positive for Type VII. and negative for Type II. 
r . 
- 2r - 1 r - 4. 
Efficiency 
of 
-1 2 - b 2 
r - 1 v - 2 
d F (V)- F (V)j 
- 2 r - 1 r - 4. 
Efficiency 
of r, A . 
2 
4 
0 
4 
0 
3 
4-93480 
0-0576 
5-1595 
0-0431 
4 
5-15947 
0-2056 
5-5648 
0-1445 
5 
5-23966 
0-3590 
5-7410 
0-2613 
6 
5-27578 
0-4865 
5-8305 
0-3708 
7 
5-29472 
0-5857 
5-8813 
0-4653 
8 
5-30576 
0-6615 
5-9126 
0-5441 
9 
5-31271 
0-7198 
5-9331 
0-6090 
10 
5-31736 
O-7650 
5-9473 
0-6624 
11 
5-32060 
0-8005 
5-9574 
0-7063 
12 
5-32296 
0-8287 
5-9649 
0-7427 
13 
5-32472 
0-8516 
5-9706 
0-7731 
14 
5-32607 
0-8702 
5-9750 
0-7986 
15 
5-9787 
0-8202 
In both cases the region of validity is bounded by the rectangle, at the point B 
(fig. 2, p. 343). Efficiency of 80 per cent, is reached when r is about 14-1 — 2-65). 
Thus for symmetrical curves of the Pearsonian type we may say that the method of 
moments has an efficiency of 80 per cent, or more, when /3 2 lies between 2*65 and 3-42. 
The limits within which the values of the parameters obtained by moments cannot be 
greatly improved are thus much narrower than has been imagined. 
11. The Reason for the Efficiency of the Method of Moments in a Small 
Region surrounding the Normal Curve. 
We have seen that the method of moments applied in fitting Pearsonian curves has 
an efficiency exceeding 80 per cent, only in the restricted region for which /3 2 lies between 
the limits 2*65 and 3-42, and as we have seen in Section 8, for which /3 , does not exceed 
0 • 1. The contours of equal efficiency are nearly circular or elliptical within these 
limits, if the curves are represented as in fig. 2, p. 343, and are ultimately centred round 
the normal point, at which point the efficiencies of all parameters tend to 100 per cent. 
It was, of course, to be expected that the first two moments would have 100 per cent, 
efficiencies at this point, for they happen to be the optimum statistics for fitting 
the normal curve. That the moment coefficients /3 X and also tend to 100 per cent, 
efficiency in this region suggests that in the immediate neighbourhood of the normal 
