Page 254 - Contributed Paper Session (CPS) - Volume 1

CPS1304 Manik A.
⁄
ℎ , h is a multiple of s,
(ℎ) = {
0, otherwise.

The k-step ahead conditional pmf (analogous to GINAR(1)) is

(1 − − + ) − ∑ ( ) − (1 − )

−

( + = y| − = x) = =0 (4)
+ ( ) (+1) (1 − ) , ≤ ,
−

{(1 − − + ) − ( + (1 − )) , > .

where, = [ ] . The k-step ahead conditional mean and variance are

respectively,

( + | − ) = − + (1 − ) (5)
1 −
and

2
)
( + | − = (1 − ) − + ( (1 − −1 − + 2−1 )) + 1− 2 , (6)
1 − 2 1− 2
where,
(1−) (1−)(1+)
2
= 1− and = (1−) 2 ,

are the mean and variance of From the equations (5) and (6), we observe

that, as → ∞ the conditional mean and variance converge to the marginal
mean and variance respectively.

3. Estimation of the parameters of GINAR(1)s model
In this section we consider the maximum likelihood and conditional least
squares estimation of the model parameters. Conditional maximum likelihood
estimators can be obtained by maximizing the conditional log-likelihood
function

log ( , … , ; , ) = ∑ log (X |X − ),

1

=+1

where, (X |X − ) is given in (2). The conditional least squares estimates of the

parameters can be obtained by minimizing the function

2
(, ) = ∑ (X − (X |X − ))

=+1
with respect to and . The differentiation results in the following estimating
equations,

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