Bee. Ho2t+b) Jeotb) Mithi 
= (5.169) 
“(i -wey | 1-wF 1 -py 
and 
2 D, 
2 
l—-uy 1l-wy 1-fip2 
2 2 
ig ee Ce. Laney , 2+ bit Pi)\ 65 170) 
(@1-H2)) | A-#y) (1-47) 1—p1f2 
_ 9 fh (uy +b)? (uy + bi) + bi) 
1) S © ee ey ee 
(41 — 2) (1-47) 1-Myl2 
2 
ace L2 : ey) — Gt bia + 51) (5.171) 
(4 — 2) (1-45) 1-12 
Using the B), Bz in Eq. 5.166, we get the general expression for R(r). 
The autocovariance function can be expressed using Green’s function as Eq. 5.149. 
Then, starting from the expression of Green’s function 
Gj = 81 Mit 82 M3, 
the same results can be derived for R(r), although the manipulations will not be shown 
here. 
52.4.5 Estimations of a,, a2, b,, and oz of ARMA(2.1). From Eq. 5.157 
R(r) = —a,R(r—- 1) —a2R(r - 2) = 2. 
Therefore, for estimation, with the sample values of autocovariances, 
R(2) = —4,R(1) — 42R(O) (5.172) 
R(3) = —4,R(2) —4>R(A). (5.173) 
143 
