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1 Stable distributions
1.5.4 Maximum likelihood method
The maximum likelihood (ML) estimation scheme for α-stable distributions
does not differ from that for other laws, at least as far as the theory is concerned.
For a vector of observations x = (x1, ..., xn), the ML estimate of the parameter
vector θ = (α, σ, β, μ) is obtained by maximizing the log-likelihood function:
n
Lθ ( χ ) = X in f( χi ; θ ), (1∙16)
i=1
where f(∙; θ) is the stable pdf. The tilde denotes the fact that, in general,
we do not know the explicit form of the density and have to approximate it
numerically. The ML methods proposed in the literature differ in the choice of
the approximating algorithm. However, all of them have an appealing common
feature - under certain regularity conditions the maximum likelihood estimator
is asymptotically normal.
Modern ML estimation techniques either utilize the FFT approach for approxi-
mating the stable pdf (Mittnik et al., 1999) or use the direct integration method
(Nolan, 2001). Both approaches are comparable in terms of efficiency. The dif-
ferences in performance are the result of different approximation algorithms,
see Section 1.3.2.
Simulation studies suggest that out of the five described techniques the method
of moments yields the worst estimates, well outside any admissible error range
(Stoyanov and Racheva-Iotova, 2004; Weron, 2004). McCulloch’s method comes
in next with acceptable results and computational time significantly lower than
the regression approaches. On the other hand, both the Koutrouvelis and the
Kogon-Williams implementations yield good estimators with the latter per-
forming considerably faster, but slightly less accurate. Finally, the ML esti-
mates are almost always the most accurate, in particular, with respect to the
skewness parameter. However, as we have already said, maximum likelihood
estimation techniques are certainly the slowest of all the discussed methods.
For example, ML estimation for a sample of a few thousand observations us-
ing a gradient search routine which utilizes the direct integration method is
slower by 4 orders of magnitude than the Kogon-Williams algorithm, i.e. a few
minutes compared to a few hundredths of a second on a fast PC! Clearly, the
higher accuracy does not justify the application of ML estimation in many real
life problems, especially when calculations are to be performed on-line.