Page 160 - Special Topic Session (STS) - Volume 2

STS474 Yoshihiro Y.
3. Result
Assumption 2 ̃ , = , , ̃ , = , .

(1) , → 0 ̃ , → ∞ → ∞.

(2) log = (log ( )) (̃ , , + |log( ⁄ ))| + |log( , ⁄ ))|)/
,
,
,
, → 0 as → ∞.
(3) ≥ 2, → ∞.

−1
−2
2
Assumption 3 , , , − ℎ ̃ , → ∞.
Then we have the following asymptotic properties of the estimator.
̂
Theorem 1 Under Assumption 1 and 2, converges to in probability as

0
→ 1.
1
̂
Theorem 2 Under Assumptions 1-3, for ≥ 2, 2( − ) converges to
0

(0, 1) in distribution as → ∞.

4. Discussion and Conclusion
This section examines empirical properties of the Gaussian semiparametric
estimation (GSE) estimators for the parameter in comparisons with those of
Zhu& Stein (2002), which is a spatial domain method that assumes that in
0
Assumption 1 is a constant. Our interests are in the comparisons between
them when is not a constant.
0
To conduct the comparisons, we simulate spatial data on lattice points that
satisfies Assumption 1 in the following three cases. In Case 1, we considered
FBFs with = 0.5, denoted as (, ). Case 1 clearly satisfies Assumption 1
0.5
with being a constant. In Case 2, for iid noise (, ), we generate FBF
0
contaminated with the noise, which is given by (, ) + (, ). In Case 3,
0.5
we simulate the moving average of (, ),given by
0.5
(1 + 0.3 )(1 + 0.3 ) (, ),
2
0.5
1
for the backward shift operators defined by (, ) = ( − 1, ) and
1
(, ) = (, − 1).
2
We simulated 100 sets of Cases 1, 2 and 3, for which we constructed the
two kinds of estimators to examine the empirical comparisons between them.
The sample paths of cross section over s = 1 for the three cases are shown in
Figure 1. It should be noticed that the each of three sample paths is a
realization of ISRF with = 0.5,
We calculated the two kinds of estimators by GSE and Zhu& Stein (2002)
for the three cases, where we chose the filter 1 with = 2 for = 1, 2, 3, 4, 5
to conduct OLS for the latter estimator.

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