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 consideration is indeed -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
-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
-stable laws.
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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 yields
the estimate of the tail index
, through the relation
.
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 may indicate a non-
-stable power-law
decay in the tails or the contrary - an
-stable
distribution with
. This is illustrated in Figure 1.4. In the left panel a power-law fit to the tail of a sample of
standard symmetric (
,
)
-stable distributed variables with
yields an estimate of
. However, when the sample size is increased to
the power-law fit to the extreme tail observations yields
, which is fairly close to the original value of
.
The true tail behavior (1.1) is observed only for very large (also for very small, i.e. the negative tail) observations,
after a crossover from a temporary power-like decay (which surprisingly indicates
). Moreover, the obtained estimates still have a slight positive bias, which
suggests that perhaps even larger samples than
observations
should be used. In Figure 1.4 we used only the
upper 0.15% of the records to estimate the true tail exponent. In
general, the choice of the observations used in the regression is
subjective and can yield large estimation errors, a fact which is often
neglected in the literature.
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A well known method for estimating the tail index that does not
assume a parametric form for the entire distribution function, but
focuses only on the tail behavior was proposed by
Hill (1975). The Hill estimator is used to estimate the
tail index , when the upper (or lower) tail of the
distribution is of the form:
,
see Figure 1.5. Like the
log-log regression method, the Hill estimator tends to
overestimate the tail exponent of the stable distribution if
is close to two and the sample size is not very large. For a review of the extreme value
theory and the Hill estimator see Härdle, Klinke, and Müller (2000, Chapter 13) or Embrechts, Klüppelberg, and Mikosch (1997).
These examples clearly illustrate that the true tail behavior of
-stable laws is visible only for extremely large data
sets. In practice, this means that in order to estimate
we must use high-frequency data and restrict ourselves to
the most ``outlying'' observations. Otherwise, inference of the tail
index may be strongly misleading and rejection of the
-stable regime unfounded.
We now turn to the problem of parameter estimation. We start the discussion with the simplest, fastest and ... least accurate quantile methods, then develop the slower, yet much more accurate sample characteristic function methods and, finally, conclude with the slowest but most accurate maximum likelihood approach.
Given a sample
of independent and identically distributed
observations, in what follows, we provide estimates
,
,
, and
of all four stable law parameters.
Already in 1971 Fama and Roll provided very simple estimates for parameters of symmetric (
) stable laws when
.
McCulloch (1986) generalized and improved their method. He analyzed stable law quantiles and provided consistent estimators of all four stable parameters, with the restriction
, while retaining the computational simplicity of Fama and Roll's method. After McCulloch define:
Statistics and
are functions of
and
. This relationship may be inverted and the parameters
and
may be viewed as functions of
and
:
Scale and location parameters, and
, can be
estimated in a similar way. However, due to the discontinuity of
the characteristic function for
and
in
representation (1.2), this procedure is much more
complicated. We refer the interested reader to the original work
of McCulloch (1986).
Given a sample
of independent and identically distributed (i.i.d.) random variables, define the sample characteristic function by
Press (1972) proposed a simple estimation method, called
the method of moments, based on transformations of the
characteristic function. The obtained estimators are consistent since
they are based upon estimators of ,
and
, which are known to be consistent. However,
convergence to the population values depends on a choice of four points at which the above functions 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 prespecified
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
. Typically no more than two or three iterations are
needed. The speed of the convergence, however, depends on the
initial estimates and the convergence criterion.
The regression method is based on the following observations
concerning the characteristic function . First, from
(1.2) we can easily derive:
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Once
and
have been obtained and
and
have been fixed at these values, estimates of
and
can be obtained using (1.14). Next, the
regressions are repeated with
,
,
and
as the initial parameters. The
iterations continue until a prespecified convergence criterion is
satisfied.
Kogon and Williams (1998) eliminated this iteration procedure and
simplified the regression method. For initial estimation they
applied McCulloch's (1986) method, worked with the continuous
representation (1.3) of the characteristic function
instead of the classical one (1.2) and used a fixed
set of only 10 equally spaced frequency points . In terms of
computational speed their method compares favorably to the
original method of Koutrouvelis (1980). It has
a significantly better performance near
and
due to the
elimination of discontinuity of the characteristic function.
However, it returns slightly worse results for very small
.
The maximum likelihood (ML) estimation scheme for -stable distributions does not differ from that for other laws, at least as far as the theory is concerned. For a vector of observations
, the ML estimate of the parameter vector
is obtained by maximizing the log-likelihood function:
Modern ML estimation techniques either utilize the FFT-based approach for approximating the stable pdf (Mittnik et al.; 1999) or use the direct integration method (Nolan; 2001). Both approaches are comparable in terms of efficiency. The differences in performance result from different approximation algorithms, see Section 1.2.2.
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 performing considerably faster, but slightly less accurate. Finally, the ML estimates 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 using 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.