Page 95 - Textos de Matemática Vol. 47

PERIODIC THRESHOLD PROCESSES 85
Note that the model in (1.1) can be represented as
Xt = t Xt 1 + Zt, (1.4)
where Zt = Z(1)I(1) + Z(2)I(2), t = ↵j ⌘ ↵(1)I(1) + ↵(2)I(2), such as ↵j 2 jj jj jj jj
(0, 1), t = j + sT , j = 1, . . . , T , and s 2 N. Furthermore, the thinning operator is defined as
and
B
Mt :=B Zt =B T 1 2
.
... ... ... . C B Z2+tT C
Xt 1 Xt 1
t Xt 1 =d X Ui,t(↵(1))I(1) + X Ui,t(↵(2))I(2)
B Q ↵T 1 i Bi=0
@ T 1 1
T 1 3 C B@ . CA. Q ↵T 1 i ... 1 0C .
Q ↵T i i=0
Q ↵T i ... ↵T 1 i=0
jj jj i=1 i=1
with (Ui,t(↵(1))) and (Ui,t(↵(2))), i 2 N, being periodic sequences of i.i.d. jj
Bernoulli r.v’s with success probabilities P(Ui,t(↵(k)) = 1) = ↵(k), for k 2 jj
{1,2}. Moreover, the innovation process (Zt) forms a periodic sequence of in- dependent Poisson-distributed r.v’s with mean vt, Zt ⇠ P o(vt), where vt = j , t = j +sT, j = 1,...,T, s 2 N0. It is assumed that Zt is independent of Xt 1 and ↵t Xt 1, for every t.
The PSETINAR(2; 1, 1)T process in (1.1) can be embedded in the following vectorial form
Yt = A Yt 1 + Mt,
being Yt = [X1+tT X2+tT · · · XT +tT ]0, where 0 denotes matrix transpose,
(1.5)
000... ↵1 1 B 0 0 ... ↵1↵2 C
A=B . ... ...
B@ Q CA
. C 00... ↵T j
T 1 j=0
010...001
B ↵2 1 ...00C0 1
B ↵3↵2
↵3 ... 0 0 C Z1+tT
i=0 C ZT+tT T 1 2 A