, p q, , .. σ2 . , . , , , , , . .
, , , , . , (), .
AR (p)
AR (p), (2.1), -. AR (p) σ2. ( + 1) . -
(2.35)
Γ - ; γ = [γ(1),..., γ(p)] T.
γ(j), j = 0,..., p -
(2.36)
(2.35) .
, > 0, m = 1, 2,..., (2.36)
(2.37)
(2.38)
- * .
, AR ( 1) AR (2) - (2.37)
,
- j - .
.
, , ( ) , , , , .
, -, , -. AR () [4]
|
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n - .
(q)
(q) . () (q) [1]:
1. , at = 0 t ≤ 0. , , , , a 1 = Xt a 2 = X 2 - θ1 a 1. , -.
2. at, t ≤ 0 , . . -, , -, . - .
(1),
Xt = at + θ at -1, t = 1,, T.
x 1,..., xT a 1,..., aT a 0 ()
a 1 = X 1 θ a 0,
a 2 = X 2 θ a 1 = X 2 θ(X 1 θ a 0),
...
a T = XT θ aT 1.
θ , x 1,..., xT a 0,
Q (θ) = a 12 + a 22 +... + aT 2,
θ. (q). , , , θ.
, q (q) , (2.18). ρ1, ρ2,..., ρ q θ1, θ2,.., θ q q q . θ , (2.18) . ( ) θ a 0, . , θ < 1, θ 1.
, a 0 = 0. a 0 , , a 0 = 0, a 0 . MA (q) q > 1: , a 0 = a -1 =... = a - q +1 = 0. (-), a 0, a -1,..., a - q +1.
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ARMA (p, q)
, . p + q + 1
. c . (2.29),
, . .
, , θ.
ARMA (p, q) [2,7]. . , , , ( , ) .
, , , .
, . . , { at } . , at - , . at. , , .
( /σ a 2) .
. , () , . , . , , , () .