Stable Distributions

1.5 Estimation of parameters

if α 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 Hardle, Klinke, and Müller
(2000, Chapter 13) or Embrechts, Klüppelberg, and Mikosch (1997).

These examples clearly illustrate that the true tail behavior of α-stable laws
is visible only for extremely large data sets. In practice, this means that in
order to estimate α 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 α-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 x₁, ..., x_n of independent and identically distributed
S_α(σ,β,μ) observations, in what follows, we provide estimates <^, σ, β, and μ
of all four stable law parameters.

1.5.2 Quantile estimation

Already in 1971 Fama and Roll provided very simple estimates for parame-
ters of symmetric (β = 0,μ = 0) stable laws when α > 1. McCulloch (1986)
generalized and improved their method. He analyzed stable law quantiles and
provided consistent estimators of all four stable parameters, with the restric-
tion α ≥ 0.6, while retaining the computational simplicity of Fama and Roll’s
method. After McCulloch define:

^x0.95 ^{- x}0.05

Vα = -------------,

^x0.75 ^{- x}0.25

which is independent of both σ and μ. In the above formula Xf denotes the f-th
population quantile, so that S_α(σ,β,μ)(Xf ) = f. Let v_α be the corresponding
sample value. It is a consistent estimator of vα . Now, define:
and let vβ be the corresponding sample value. Vβ is also independent of both
σ and μ. As a function of α and β it is strictly increasing in β for each α. The
statistic ^β is a consistent estimator of vβ.

Statistics vα and vβ are functions of α and β . This relationship may be inverted

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