10
1 Stable distributions
is Sα (σ, β, μ). It is interesting to note that for α = 2 (and β = 0) the Chambers-
Mallows-Stuck method reduces to the well known Box-Muller algorithm for
generating Gaussian random variables (Janicki and Weron, 1994). Although
many other approaches have been proposed in the literature, this method is
regarded as the fastest and the most accurate (Weron, 2004).
1.5 Estimation of parameters
Like simulation, the estimation of stable law parameters is in general severely
hampered by the lack of known closed-form density functions for all but a few
members of the stable family. Either the pdf has to be numerically integrated
(see the previous section) or the estimation technique has to be based on a
different characteristic of stable laws.
All presented methods work quite well assuming that the sample under con-
sideration is indeed α-stable. However, if the data comes from a different
distribution, these procedures may mislead more than the Hill and direct tail
estimation methods. Since the formal tests for assessing α-stability of a sample
are very time consuming we suggest to first apply the “visual inspection” tests
to see whether the empirical densities resemble those of α-stable laws.
1.5.1 Tail exponent estimation
The simplest and most straightforward method of estimating the tail index is
to plot the right tail of the empirical cdf on a double logarithmic paper. The
slope of the linear regression for large values of x yields the estimate of the tail
index α, through the relation α = -slope.
This method is very sensitive to the size of the sample and the choice of the
number of observations used in the regression. For example, the slope of about
-3.7 may indicate a non-α-stable power-law decay in the tails or the contrary
- an α-stable distribution with α ≈ 1.9. This is illustrated in Figure 1.4. In
the left panel a power-law fit to the tail of a sample of N = 104 standard
symmetric (β = μ = 0, σ = 1) α-stable distributed variables with α = 1.9
yields an estimate of α = 3.732. However, when the sample size is increased to
N = 106 the power-law fit to the extreme tail observations yields α = 1.9309,
which is fairly close to the original value of α.
The true tail behavior (1.1) is observed only for very large (also for very small,