Page 101 - Textos de Matemática Vol. 47

PERIODIC THRESHOLD PROCESSES 91
is the Fisher information matrix with
26 E" @2`(✓) # E" @2`(✓) # E" @2`(✓) # 37 6 @(↵(1))2 @↵(1)@↵(2) @↵(1)@ j 7
Mj = 6 E" @2`(✓) # E" @2`(✓) # E" @2`(✓) # 7,
6 @↵(1)@↵(2) @(↵(2))2 @↵(2)@ j 7 6jj j j7
jjjj
64 E " @2`(✓) # E " @2`(✓) # @ ↵ ( 1 ) @ j @ ↵ ( 2 ) @ j
for j = 1,...,T, and
E @2`(✓) 75 @ 2j
h(2↵(k) 1)b a↵(k)2i j j
E
+1+1 (k) ⇢ @(↵(k))2 ↵(k)(1 ↵(k))
jj
"#
@2`(✓) = NX XP (Xj+sT = b) Ij
jabjj
(k) (k)2 (k)
⇥pj(b|a)+2(1 ↵j ) jpj(b 1|a) + jpj(b 2|a) (k) (k) 2 (k) 2 p2j (b 1|a)(k)
+2(1 ↵j ) jpj(b 1|a) + jpj(b 2|a) j pj(b|a)(k) ; " # +1+1 ⇢
E @2`(✓) = N X XP(Xj+sT =b) pj(b 1|a)(k)
@↵(k)@ j ↵(k)(1 ↵(k)) jjjab
(k) p2(b 1|a)(k) j pj (b 2|a) + j p(b|a)(k) ;
+1+1 ( 2 (k) ) @ 2j a b p j ( b | a ) ( k )

E @2`(✓) =NXXP(Xj+sT =b) jpj(b 2|a)(k) pj(b 1|a) ,
for k = 1, 2.
Proof. In order to derive the large sample distribution of the CML-estimators, we use the same arguments as in Franke and Seligmann (1993, pp. 324–325). Note that the consistency and asymptotic distribution of the CML-estimators for the INAR(1) process can be obtained by means of Theorems 2.1 and 2.2 in Billingsley (1961, pp. 10–13). Since the innovation process is Poisson-distributed the arguments used by Monteiro et al. (2010, 2012) for the periodic INAR of order one and period T , PINAR(1)T , and the SETINAR(2; 1, 1) process can be easily generalized for the SETINAR(2; 1, 1) with periodic structure. We omit the details. ⇤