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different choices of the arguments of the characteristic functions of the underlying
stable distribution. The regression-type estimator was proposed by
Kouttrouvelis
(
1980
).
Zolotarev
(
1986
) established an estimator of the stable index
based on
special transformations in the case of strictly stable distributions.
McCulloch
(
1986
)
suggested using a quantile-based estimator.
Ma and Nikias
(
1995
)and
Tsihrintzis
and Nikias
(
1996
) presented the estimators based on moment properties of stable
random variables. In addition,
Meerschaert and Scheffer
(
1998
) constructed a robust
estimator based on the weak convergence of the distributions of partial sums.
Estimators based on the method of point process were introduced by
Marohn
(
1999
).
Nolan
(
2001
) proposed the maximum likelihood estimation for stable
parameters.
Recent work by
Fan
(
2006
) proposed an estimator for the stable index by
using the structure of U-statistics. His estimator is not only unbiased but is also
consistent. Additionally, this estimator is simple and easy-to-compute. It is well
known that V-statistics is a linear combination of U-statistics (see, for example,
Lee
1990
;
Nomachi and Yamato 2001
). Furthermore, in some cases, V-statistics has a
mean square error (MSE) smaller than U-statistics (
Shao 2003
). The asymptotic
normality of V-statistics is discussed in Chap. 6 of
Serfling
(
1980
). Therefore, we
have developed a new estimator of stable index by extending the concepts of
Fan
(
2006
), by using the structure of V-statistics.
The paper is structured as follows. In Sect.
2
, we describe stable distributions.
The parameter estimations for the stable index are shown in Sect.
3
. Section
4
presents the results of simulations. Conclusions are presented in the final section.
α
2
Stable Distributions
A nondegenerate distribution is a stable distribution if it satisfies the following
property: Suppose
X
1
and
X
2
are independent copies of a random variable
X
.Then
X
is said to be stable if for any constants
a
>
0and
b
>
0, the random variable
aX
1
+
bX
2
has the same distribution as
cX
+
d
for the constants
c
>
0and
d
.The
distribution is said to be strictly stable if this holds with
d
0(
Nolan 2009
). In
general, a stable distribution does not have closed-form expressions for its density
and distribution function. However, this distribution can be described easily by its
characteristic function. A random variable
X
is said to have a stable distribution if
the probability density has the following form:
=
∞
−
∞
φ
(
1
2
e
−
ixt
d
t
f
(
x
)=
t
)
,
(1)
π
where
exp
−
σ
α
|
|
α
[
(
i
μ
t
t
1
−
i
β
sgn
(
t
)
tan
(
πα
/
2
)])
,
if
α
=
1,
φ
(
t
)=
(2)
2
π
exp
(
i
μ
t
−
σ
|
t
|
[
1
−
i
β
sgn
(
t
)
log
|
t
|
)
,
if
α
=
1,
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