Stable laws - also called -stable, stable Paretian or Lévy stable
- were introduced by Levy (1925) during his investigations of the behavior of sums of independent random variables.
A sum of two independent random variables having an
-stable distribution
with index
is again
-stable with the same index
. This
invariance property, however, does not hold for different
's.
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The -stable distribution requires four parameters for
complete description: an index of stability
also
called the tail index, tail exponent or characteristic exponent, a
skewness parameter
, a scale parameter
and a location parameter
. The tail
exponent
determines the rate at which the tails of the
distribution taper off, see the left panel in Figure 1.1.
When
, the Gaussian distribution results. When
, the variance is infinite and the tails are asymptotically equivalent to a Pareto law, i.e. they exhibit a power-law behavior. More precisely, using a central limit theorem type argument it can be shown that (Samorodnitsky and Taqqu; 1994; Janicki and Weron; 1994):
When , the mean of the distribution exists and is equal to
. In general, the
th moment of a stable random variable is finite if and only if
. When the skewness parameter
is positive, the
distribution is skewed to the right, i.e. the right tail is
thicker, see the left panel of Figure 1.2. When it is negative, it
is skewed to the left. When
, the distribution is
symmetric about
. As
approaches 2,
loses its
effect and the distribution approaches the Gaussian distribution
regardless of
. The last two parameters,
and
, are the usual scale and location parameters, i.e.
determines the width and
the shift of the mode (the peak) of
the density. For
and
the distribution is called standard stable.
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Due to the lack of closed form formulas for densities for all but
three distributions (see the right panel in Figure 1.2), the -stable law can be most conveniently described by its characteristic function
-
the inverse Fourier transform of the probability density function.
However, there are multiple parameterizations for
-stable
laws and much confusion has been caused by these different
representations, see Figure 1.3. The variety of formulas is caused by a
combination of historical evolution and the numerous problems that
have been analyzed using specialized forms of the stable
distributions. The most popular parameterization of the
characteristic function of
,
i.e. an
-stable random variable with parameters
,
,
, and
, is given by (Weron; 2004; Samorodnitsky and Taqqu; 1994):
The lack of closed form formulas for most stable densities and distribution functions has negative consequences. For example, during maximum likelihood estimation computationally burdensome numerical approximations have to be used. There generally are two approaches to this problem. Either the fast Fourier transform (FFT) has to be applied to the characteristic function (Mittnik, Doganoglu, and Chenyao; 1999) or direct numerical integration has to be utilized (Nolan; 1997, 1999).
For data points falling between the equally spaced FFT grid nodes an interpolation technique has to be used. Taking a larger number of grid points increases accuracy, however, at the expense of higher computational burden. The FFT based approach is faster for large samples, whereas the direct integration method favors small data sets since it can be computed at any arbitrarily chosen point. Mittnik, Doganoglu, and Chenyao (1999) report that for the FFT based method is faster for samples exceeding 100 observations and slower for smaller data sets.
Moreover, the FFT based approach is less universal - it is efficient only for large
's and only for pdf calculations. When computing the cdf the density must be numerically integrated. In contrast, in the direct integration method Zolotarev's (1986) formulas either for the density or the distribution function are numerically integrated.
Set
. Then the density
of a standard
-stable random variable in representation
, i.e.
, can be expressed as (note, that Zolotarev (1986, Section 2.2) used yet another parametrization):
where
and
The distribution
of a standard
-stable random variable in representation
can be expressed as:
Formula (1.5) requires numerical integration of the function
, where
. The integrand is 0 at
, increases monotonically to a maximum of
at point
for which
, and then decreases monotonically to 0 at
(Nolan; 1997). However, in some cases the integrand becomes very peaked and numerical algorithms can miss the spike and underestimate the integral. To avoid this problem we need to find the argument
of the peak numerically and compute the integral as a sum of two integrals: one from
to
and the other from
to
.