Civil Engineering Reference
In-Depth Information
Hence,
n
−
1
1
n
.
V
n
(
h
)=
U
n
(
h
)+
(8)
n
is therefore
V
−
1
n
The alternative estimator for stable index
α
(
h
)
defined in Eqs. (
7
)
and (
8
).
4
Simulation Results
In this section, we present the simulation results in order to examine the efficiency of
the estimators,
U
−
1
n
and
V
−
1
n
(
h
)
(
h
)
, of the stable index
α
. Simulation experiments
are conducted with a fix location parameter
μ
=
0, the skewness parameter
β
=
0,
the scale parameter
1, and the stable parameter ranging from 0.1 to 1.9. All
data were simulated by using R statistical software (
Ihaka and Gentleman 1996
).
The number of simulation trials was set to 1,000 and the sample sizes simulated
were 200 and 500. We used a function
rstable
from package Basics so as to
generate samples from corresponding stable distributions. Simulation results which
are shown in Tables
1
and
2
consist of mean, standard deviation, interquartile
range (IQR), and MSE of the estimators. As can be seen from Table
1
and Fig.
2
,
the standard deviation and IQR of our estimator
V
−
1
n
σ
=
(
h
)
are smaller than those of the
estimator
U
−
1
n
5 and the MSE of the estimator
V
−
1
n
(
h
)
when
α
≥
0
.
(
h
)
are smaller
than that of the estimator
U
−
1
n
5, the two estimators
have very similar standard deviation, IQR, and MSE. These values decrease as
sample sizes get larger. As it is seen from Fig.
2
, the standard deviations, IQRs,
and MSEs of estimators increase when the values of
(
h
)
when
α
≥
0
.
6. When
α
<
0
.
α
increase. Additionally, the
standard deviation, IQR, and MSE of the estimator
V
−
1
n
(
h
)
are less than those of the
estimator
U
−
1
n
(about 0.01-1.6 %, 0.004-1.4 %, and 0.03-3.8 %, respectively).
The simulation results for
n
(
h
)
500 shown in Table
2
are similar to those reported in
Tab le
1
. Our empirical evidence indicates that the estimator
V
−
1
n
=
(
h
)
performs better
than the estimator
U
−
1
n
in terms of the standard deviation, IQR, and MSE for
almost all scenarios that we considered.
(
h
)
5
Conclusions
In this paper, we propose a new estimator of stable index for the stable distribution
based on the structure of V-statistics. The proposed estimator,
V
−
1
n
(
h
)
,andthe
existing estimator based on U-statistics,
U
−
1
n
, were compared through a simu-
lation study. Simulation results show that the estimator
V
−
1
n
(
h
)
(
h
)
is more efficient
than the estimator
U
−
1
n
in terms of standard deviation, IQR, and MSE in almost
all situations. In addition, the comparative efficiency increases as
(
h
)
α
approaches 2.
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