1.4 Simulation of α-stable variables
1.4 Simulation of α-stable variables
The complexity of the problem of simulating sequences of α-stable random
variables results from the fact that there are no analytic expressions for the
inverse F-1 of the cumulative distribution function. The first breakthrough
was made by Kanter (1975), who gave a direct method for simulating Sα (1, 1, 0)
random variables, for α < 1. It turned out that this method could be easily
adapted to the general case. Chambers, Mallows, and Stuck (1976) were the
first to give the formulas.
The algorithm for constructing a standard stable random variable X ~ Sα(1,β,0),
in representation (1.2), is the following (Weron, 1996):
• generate a random variable V uniformly distributed on ( —π, π ) and an
independent exponential random variable W with mean 1;
• for α 6= 1 compute:
|
si{ s sin{α(v + Bα,β)} α,β ∙ {cos(V)}1 /α ' |
cos{V — α ( V + Bα,β ) } . W . |
(1 -α )/α , |
(1.6) |
|
where |
arctan(β tan πα )
Bα,β = ---------------2 ,
α
2 2 παa 1 / (2α)
Sα,β = 1 + β2 tan2 Ç—) I
• for α = 1 compute:
X=
πW cos Vλ ɪ
∏ + βv Л
(1.7)
Given the formulas for simulation of a standard α-stable random variable, we
can easily simulate a stable random variable for all admissible values of the
parameters α, σ, β and μ using the following property: if X ~ Sα (1 ,β, 0) then
{σX + μ, α = 1,
(1.8)
σX + 2 βσ ln σ + μ, α = 1,