R(j)+a,RG-1)+- - -ajR)- - -+a,RG—n) = 0 (5.239) 
PS ilo o om 
From this relation d;,@2,- - -d;- - -d, can be estimated by solving the simultaneous 
equations 
a,R(0) + a>R(1) +: - -a,R(n— 1) =—R(1) 
a,R(1) + aoR(0)+- - -+a,R(n—2) =—R(2) 
(5.240) 
a,R(n—1)+aR(n—2)+- - -+a,R(O) = —R(n). 
This can be transformed, using the vector expression, to 
(a1,02,+ + +a)’ =a@ (5.241) 
{ia Vio <5 Ro} =e, (5.242) 
RO) R()--:R(n-1) 
RG) (RO RO 2) 
R,= |. ne (5.243) 
R(n-1) R(n-2) R(0) 
R,-a=-r. (5.244) 
Therefore 
A -—Tr 
OF aaa (5.245) 
Rn 
Equation 5.239 or 5.240 is identical with the Yule-Walker equations. 
All these relations mean that, if the approximations as given here are introduced, the 
maximum likelihood method can be replaced by the least squares method, and the least 
squares method is reduced to get the solution of the Yule—-Walker equations. Priestley”? 
describes in more detail the precise estimation through a rigorous likelihood function. 
192 
