Civil Engineering Reference
In-Depth Information
X
n
are independent and identically distributed (iid) random
variables with strictly stable distribution. Based on the sum-preserving property of
stable random variables [see Definition 2.2 in
Fan
(
2006
)], we have
Suppose
X
1
,...,
d
=
2
1
/
α
·
X
1
+
X
2
X
1
.
(3)
Further,
1
α
E
[
log
|
X
1
+
X
2
|
]=
log 2
+
E
[
log
|
X
1
|
]
,
that is,
1
α
=
E
[
log
|
X
1
+
X
2
|
]
−
E
[
log
|
X
1
|
]
.
(4)
log 2
Let the kernel
h
be a real function given by
log
1
2
|
x
1
+
x
2
|−
(
log
|
x
1
|
+
log
|
x
2
|
)
h
(
x
1
,
x
2
)=
.
log 2
By using the kernel
h
, we define U-statistics (
Lee 1990
;
Serfling 1980
)
n
2
−
1
∑
U
n
(
h
)=
h
(
X
i
,
X
j
)
.
(5)
1
≤
i
<
j
≤
n
(
)
Fan
(
2006
)alsohasshownthat
U
n
h
defined in Eq. (
5
) is an unbiased estimator
α
−
1
. Consequently, the unbiased estimator for stable index
of parameter
α
is
U
−
1
α
−
1
was also discussed
in Theorem 2.1 of
Fan
(
2006
). Compared with the estimator of
Press
(
1972
)via
simulations, his estimator yields smaller MSE. Apart from U-statistics, there is a
closely related V-statistics defined by
(
)
h
. The asymptotic normality of the estimator of
n
n
i
=
1
n
j
=
1
h
(
X
i
,
X
j
)
.
1
n
2
V
n
(
h
)=
(6)
Although V-statistics is biased, the bias is small asymptotically. However,
V
n
(
h
)
may
be better than
U
n
(
in terms of their MSE (
Shao 2003
). In case where the kernel
h
is not degenerate, V-statistics has asymptotic normality (
Nomachi and Yamato
2001
). Therefore, in this paper we will examine the efficiency of the estimator of
h
)
α
based on V-statistics via simulations. In addition, the formula given in Eq. (
6
) can
be written as a linear combination of U-statistics (
Shao 2003
):
n
2
n
2
−
1
U
n
(
1
n
.
V
n
(
h
)=
h
)+
(7)
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