A Simulation Study of Estimator for the Stable Index - Innovative Management in Information and Production - page 10

alpha=0.5
alpha=1.0
alpha=1.9

-4 -2 0 2 4

x

Fig. 1 Stable densities S α ( 1 , 0 , 0 ) ,for α = 0 . 5 , 1 , and 1 . 9

with is the sign function of t .Since

Eq. ( 2 ) is characterized by four parameters, we will denote stable distributions by
S α ( σ , β , μ )

α ∈ ( 0 , 2 ) , β ∈ [ − 1 , 1 ] , σ > 0, μ ∈ R ,andsgn ( t )

is referred to as the stable index or characteristic
exponent of the stable distribution,

. The parameter α

σ is the scale parameter, β is the skewness

parameter, and is the location parameter. There are three special cases of stable

distribution which one can write down in closed form for the probability density
function. First, the case where

μ

α = 2 yields a normal distribution. Second, the case

where α = 1and β = 0 yields a Cauchy distribution. The last special case is obtained

for 1. For this case, we have a Levy distribution ( Nolan 2009 ).

Figure 1 displays the probability densities of stable distribution where

α = 1 / 2and β =

σ = 1, β = 0,

μ = 0, and α = 0 . 5 , 1 , 1 . 9.

When the parameter β is zero, the distribution is symmetric around μ .Stable

distribution allows for skewed distributions when 0 and the distribution is fat-

tailed ( Rachev et al. 2005 ). Two conditions yield a strictly stable distribution: when
(i)

β =

α = 1, μ = 0 and (ii) α = 1, β = 0.

3 Parameter Estimations for the Stable Index

Fan ( 2006 ) proposed an unbiased estimator for the stable index based on

U-statistics first introduced by Hoeffding ( 1948 ). In what follows, we review this
estimator which proposed by Fan ( 2006 ) and then we augment his estimator by
using V-statistics.

α