Biomedical Engineering Reference
In-Depth Information
TABLE 9.2: Monte Carlo Simulation Results of the B-Splines Sieve MLE of
0
with 1,000 Repetitions for Semiparametric Analysis of Panel Count Data
n = 50 n = 100 n = 200
0;1
0;2
0;3
0;1
0;2
0;3
0;1
0;2
0;3
Bias
0.0001
-0.0003
0.0014
0.0003
0.0005
0.0001
-0.0012
0.0003
-0.0002
M-C sd
0.1029
0.0286
0.0712
0.0685
0.0188
0.0488
0.0474
0.0141
0.0337
ASE
0.1365
0.0418
0.0865
0.0805
0.0239
0.0542
0.0519
0.0152
0.0359
95%-CP
98.4%
98.5%
97.5%
97.6%
97.9%
96.5%
96.2%
95.1%
96.6%
as sample size increases to 200. With sample size 200, the Wald 95% confidence
interval achieves the desired coverage probability.
Simulation 2: Panel Count Data. We generate the data with the setting
given in Wellner and Zhang (2007). For each subject, we independently gen-
erate X
i
= (Z
i
;K
i
;
T
(i)
(i)
K
i
), for i = 1; 2; ;n, where Z
i
= (Z
i;1
;Z
i;2
;Z
i;3
)
with Zi,3
i;1
Unif(0; 1), Z
i;2
N(0; 1), and Z
i;3
Bernoulli(0:5); K
i
is
sampled randomly from the discrete set f1; 2; 3; 4; 5; 6g; Given K
i
,
T
(i
K
i
K
i
;N
=
(T
(i)
K
i
;1
;T
(i)
K
i
;2
; ;T
(i)
K
i
;K
i
) are the order statistics of Ki
i
random draws from
Unif(0; 1). The panel countsN
(i)
K
i
;K
i
) are gener-
ated from the Poisson process with the conditional mean function given by
(tjZ
i
) = 2t exp(
0
Z
i
) with
0
= (1:0; 0:5; 1:5)
T
:
We conduct the simulation study for n = 50, 100 and 200, respectively.
(i)
K
i
(i)
K
i
;1
;N
(i)
K
i
;2
; ;N
= (N
In each case, we perform a Monte Carlo study with 1,000 repetitions. Table
9.2 displays the estimation bias (Bias), Monte Carlo standard deviation (M-C
sd), the average of standard errors using the proposed method (ASE), and the
coverage probability of the 95% Wald confidence interval using the proposed
estimator of standard error (95%-PC).
The results show that the B-splines sieve MLE performs quite well. It has
very little bias, with seemingly decreased estimation variability as sample size
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