3.6 Simulation Results
In this section, we carry out simulations to check the performance
of the proposed OPG method and MAVE method. Some comparisons of
the methods are made. Let
be the space spanned
by the column vectors in
. To describe the error of
estimation, we define the distance from
to
as
, where the columns
of
form an orthogonal standard basis of the space. It is
easy to see that
. If
then
; if
then
.
Example 3.6.1. Consider the following model
 |
|
|
(3.39) |
where
and
and
they are independent. In model (3.39), the coefficients
,
,
,
and there
are four e.d.r. directions. Let
. In our simulations, the SIR
method and the ADE method perform quite poorly for this model.
Next, we use this model to check the OPG method and the MAVE
method.
Table:
Average distance
for model (3.39) using different methods
|
|
 |
Freq. of Est. No. of |
n |
Methods |
 |
 |
 |
 |
e.d.r directions |
|
pHd |
.2769 |
.2992 |
.4544 |
.5818 |
=0 |
=10 |
=23 |
|
OPG |
.1524 |
.2438 |
.3444 |
.4886 |
=78 |
=44 |
=32 |
100 |
MAVE |
.1364 |
.1870 |
.2165 |
.3395 |
=11 |
=1 |
=1 |
|
rMAVE |
.1137 |
.1397 |
.1848 |
.3356 |
=0 |
|
|
|
pHd |
.1684 |
.1892 |
.3917 |
.6006 |
=0 |
=0 |
=5 |
|
OPG |
.0713 |
.1013 |
.1349 |
.2604 |
=121 |
=50 |
=16 |
200 |
MAVE |
.0710 |
.0810 |
.0752 |
.1093 |
=8 |
=0 |
=0 |
|
rMAVE |
.0469 |
.0464 |
.0437 |
.0609 |
=0 |
|
|
|
pHd |
.0961 |
.1151 |
.3559 |
.6020 |
=0 |
=0 |
=0 |
|
OPG |
.0286 |
.0388 |
.0448 |
.0565 |
=188 |
=16 |
=6 |
400 |
MAVE |
.0300 |
.0344 |
.0292 |
.0303 |
=0 |
=0 |
=0 |
|
rMAVE |
.0170 |
.0119 |
.0116 |
.0115 |
=0 |
|
|
|
With sample sizes
,
and
, 200 independent
samples are drawn. The average distance from the estimated e.d.r. directions
to
is calculated for the pHd
method (Li, 1992), the OPG method, the MAVE method and the rMAVE
method. The results are listed in Table 3.1. The proposed OPG and
MAVE methods work quite well. The results also show that the MAVE
method is better than the OPG method, while the rMAVE method
shows significant improvement over the MAVE method. Our method for
the estimation of the number of e.d.r. directions also works quite
well.
Example 3.6.2. We next consider the nonlinear time series model
 |
|
|
(3.40) |
where
,
are i.i.d.
,
,
and
. A typical data set sample with size 1000
was drawn from this model. The points
, and
are plotted in Figures
3.7 (a)-(f). As there is no discernible symmetry, the
SIR method will not be appropriate.
Figure 3.7:
A typical data set
from model (3.40). (a)-(f) are
plotted against
respectively.
|
Now, we use the OPG method and the MAVE method. The simulation
results are listed in Table 3.2. Both methods have quite small
estimation errors. As expected, the rMAVE method works better
than the MAVE method, and the MAVE method outperforms the OPG
method. The number of the e.d.r. directions is also estimated
correctly most of the time for suitable sample size.
Table:
Mean of the distance
for model (3.40) using different methods
|
|
for k = |
Freq. of Est. No. of |
 |
Method |
1 |
2 |
3 |
e.d.r. directions |
|
pHd |
.1582 |
.2742 |
.3817 |
=3 |
=73 |
|
OPG |
.0427 |
.1202 |
.2803 |
=94 |
=25 |
100 |
MAVE |
.0295 |
.1201 |
.2924 |
=4 |
=1 |
|
rMAVE |
.0096 |
.0712 |
.2003 |
|
|
|
pHd |
.1565 |
.2656 |
.3690 |
=0 |
=34 |
|
OPG |
.0117 |
.0613 |
.1170 |
=160 |
=5 |
200 |
MAVE |
.0059 |
.0399 |
.1209 |
=1 |
=0 |
|
rMAVE |
.0030 |
.0224 |
.0632 |
|
|
|
pHd |
.1619 |
.2681 |
.3710 |
=0 |
=11 |
|
OPG |
.0076 |
.0364 |
.0809 |
=185 |
=4 |
300 |
MAVE |
.0040 |
.0274 |
.0666 |
=0 |
=0 |
|
rMAVE |
.0017 |
.0106 |
.0262 |
|
|
|
Example 3.6.3. The multi-index model (3.6). For
simplicity, we discuss only the generalized partially linear
single-index models. This model was proposed by Xia, Tong, and Li (1999),
which has been found adequate for quite a lot of real data
sets. The model can be written as
 |
|
|
(3.41) |
where
and
.
Following the idea of the MAVE method, we may estimate
and
by minimizing
|
|
![$\displaystyle \sum_{j=1}^n \sum_{i=1}^n\Big [ y_i - \theta^{\top } X_i -
a_j - b_j \beta^{\top } (X_i - X_j)\Big ]^2
w_{ij},$](xegbohtmlimg1498.gif) |
|
|
|
 |
(3.42) |
Suppose that
and
constitute the
minimum point. Then we have the estimates
To illustrate the performance of the above algorithm, we further
consider the following model
 |
|
|
(3.43) |
where
and
are i.i.d.
. In this model,
and
. Let
and
. We
can estimate the model using the MAVE method without assuming its
specific form. The simulation results listed in Table
3.3 suggest that the estimation is quite successful. If
we further assume that it is a generalized partially linear
single-index model and estimate
and
by
minimizing (3.42), the estimation is much better as
shown in Table 3.3.
Table:
Mean of the estimated directions and average distance
or
in square brackets for model (3.43)
 |
|
no model specification |
model specified |
|
 |
(0.2703 -0.5147 0.8136) |
[.000329] |
(0.2678 -0.5346 0.8014) |
[.000052] |
50 |
 |
(0.0264 0.8487 0.5281) |
[.013229] |
(0.0108 0.8319 0.5513) |
[.003946] |
|
 |
(0.2679 -0.5307 0.8041) |
[.000052] |
(0.2665 -0.5346 0.8020) |
[.000019] |
100 |
 |
(0.0035 0.8341 0.5516) |
[.002244] |
(0.0014 0.8318 0.5540) |
[.001142]
|
|
Example 3.6.4. The varying-coefficient model (3.7). For
simplicity, here we only discuss the single-index coefficient
linear model proposed by Xia and Li (1999). The model can be
written as
 |
|
|
(3.44) |
where
.
To see the performance of the MAVE method discussed in Section
3.3.2, we further consider the following model
 |
|
|
(3.45) |
where
,
, and
are i.i.d.
. In this model,
. Model (3.45) is a
combination of the TAR model and the EXPAR model (cf.
Tong (1990)). Table 3.4 shows the simulation
results, which also suggest that the estimation is satisfactory.
Table 3.4:
Mean of the estimated directions and the standard deviation for model (3.45)
 |
Estimated direction |
s.d. |
= 100 |
(0.2794 0.5306 0.7895) |
[0.07896 0.0817 0.0646] |
= 200 |
(0.2637 0.5243 0.8052) |
[0.06310 0.0468 0.0303] |
|