FOUNDATIONS OF THEORETICAL STATISTICS. 
333 
and the equation to determine m is 
3L 
^ 
cm \x—m 
1 \ n 
+ - 
<(, 
- 0 
the accuracy of the value so obtained is found from the second differential. 
8 2 L 
of which the mean value is 
whence 
^ Li 
N-2 = 
cm \x—rn 
n 
d 2 {p — 1)’ 
_ << 2 (p— 1) 
n 
(0 
We now see that the efficiency of location by the method of moments is 
2z± = 1 _ _ 2 _ . 
p +1 p +1' 
Efficiencies of over 80 per cent, for location are therefore obtained if p exceeds 9 ; for 
p — 1 the efficiency of location vanishes, as in other cases where the curve makes an 
angle with the axis at the end of its range. 
Turning now to the problem of scaling, we have, by the method of moments, 
Mi = ((2 {p + l)j 
whence, knowing p, a is obtained. Since 
2 _ft-l 2 
^ P-2 M2 ? 
n 
we must have 
2 — I 2 4 + '3fi± 2 _ P + 4 2 . 
a,x 4n ' 8n 2(p+l)n 
on the other hand, from the value of L, we find the equation 
0L n r . , \ . 1 
8 a 
— = --(i>+l) + “2 S {x-m) = 0, . 
(_b ct 
(2) 
to be solved for m and a as a simultaneous equation with (1) ; whence 
8 2 L 
8m 8« 
a 
2 ’ 
and 
S = ^(i>+9 -r. s (*-*«). 
8« : 
a 
2 
3 A 
VOL. ccxxn. -A. 
