Stable Distributions

1 Stable distributions

and the parameters α and β may be viewed as functions of vα and vβ :

α = ψ1(vα,vβ), β = ψ2(vα,vβ).                  (1.11)

Substituting vα and vβ by their sample values and applying linear interpolation
between values found in tables provided by McCulloch (1986) yields estimators
α and /.

Scale and location parameters, σ and μ, can be estimated in a similar way.
However, due to the discontinuity of the characteristic function for α = 1 and
/ 6= 0 in representation (1.2), this procedure is much more complicated. We
refer the interested reader to the original work of McCulloch (1986).

1.5.3 Characteristic function approaches

Given a sample x₁, ..., x_n of independent and identically distributed (i.i.d.)
random variables, define the sample characteristic function by

1ⁿ

φ(t) = - ei^t^e^itx=.                          (1.12)

ⁿ j=1

Since ∣φ(t)| is bounded by unity all moments of φ(t) are finite and, for any
fixed t, it is the sample average of i.i.d. random variables exp(itxj ). Hence,
by the law of large numbers, φ(t) is a consistent estimator of the characteristic
function φ(t).

Press (1972) proposed a simple estimation method, called the method of mo-
ments, based on transformations of the characteristic function. The obtained
estimators are consistent since they are based upon estimators of φ(t), Im{φ(t)}
and Re{φ(t)}, which are known to be consistent. However, convergence to the
population values depends on a choice of four points at which the above func-
tions are evaluated. The optimal selection of these values is problematic and
still is an open question. The obtained estimates are of poor quality and the
method is not recommended for more than preliminary estimation.

Koutrouvelis (1980) presented a regression-type method which starts with an
initial estimate of the parameters and proceeds iteratively until some prespec-
ified convergence criterion is satisfied. Each iteration consists of two weighted
regression runs. The number of points to be used in these regressions depends
on the sample size and starting values of α. Typically no more than two or

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