70 
Choice in the Distrihiitioii of Observations 
from which we find 
a2 \{x -vY{x- If (1 - af {x-vf{x+\f (1 + af 
4(1 + v)2 ■ 8 4(1"- vf"' y 
<^-. = N 
The condition for 
IS 
»-! .r--l 
(1 + g) ^ _ (1 - a)2 
y 8 
.r = l 
Eliminating 8 we obtain for ol — ct^, the value 
{x^- 1) 2 (1 + at))^ 
(y2- 1)2 • - y 
.!•=] 
cr^, — ct:, : 
a2 (1 + a)2(3;2- 1) f (1 + av)^ (a;^ - 1) 
or 
.r-l 
<j'i ~ o'i ■ 
(v2-l)2 1(1 + a)2- 2y (1 + a2) 
+ ^ [(1 + «') +2v{l- t'2) + 2 - 5«2 + 
(1 + a)2 (cc2- 1) 
+ 2v (1 ^ ?;2) [(1 + a)2 - 2y (1 + a^)] a; + (1 + a)^ (2 - + v'') 
- 2y [(1 + a2) (2 - 5v^ + v') + (1 + avf]} (87). 
x = l 
Our assumption that the maximum cr'; shall be equal to al requires that the 
expression in curled brackets shall be a perfect square for which the condition is 
y_ ' 
+ (- 2 + 9a2 + 5a*) v2 + 2a (3 + a^) v - (3 + 2a2)} 
2 
{a2 (1 + a2) + 2a (1 + a^) + (3 - - 3a*) ?'« - 4a (3 + 2a2) 
{- a^v^ - 2av^ + (- 5 + 2a2) w* + I2av^ - (2 + 9a2) «2 _ 2av 
(l + a)2 
+ 3 + 4a2} + -y4 + 2^2 _ 1 = 0 
,'(2 
.(88). 
Now a-„ = . 
' N y 
is the maximum which we want to make as low as 
possible, hence we have for a certain a to find the v for which ^ — given by 
(88) is a maximum. 
We shall examine the cases a = -5 and a = -9. 
(10) For a = -5 (88) takes the form 
y _ 
(1 + a)2^ 
{•625u« + 2-5v5 + 5-125?;* - Mv^ + M25r2 + 6-5w - 1-75} 
7 
(1 
{-•2.5r« 
4-5?;* + 6?;« - 4-25i'2 _ „ + 4} + -i;4 + 2i)2 _ 1 = o. 
which differentiated with regard to v gives 
L(i + o.f 
+ 
{3-75-!)« + 12-5y* + 20-5w3 - 42^2 + 2-25?; + 6-5} 
{- l-5i)5 - 5d* - 18^3 + 18?;2 - 8-5v - 1} + 4v (^2 + 1) = 0. 
y 
(1 + a) 
